#72249473

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15259
commutator, Frattini, cosocle	C₂⁶	C₂⁴
prime radical	#62817	C₂²
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

3775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6,10,14)(4,11)(5,8)(7,12)(9,15,16,13)(17,18)(19,23)(20,24)(21,32,35,25)(22,26)(27,39)(28,45,33,37)(29,38)(30,40,34,42)(31,41)(36,44)(43,48,46,47);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,15)(12,16)(17,19,22,29)(18,23,26,38)(20,27,31,44)(21,33,35,28)(24,39,41,36)(25,37,32,45)(30,43,34,46)(40,47,42,48);(1,2)(3,6,10,14)(4,7,12,11)(5,8)(9,13)(15,16)(17,20,22,31)(18,24,26,41)(19,27)(21,34,35,30)(23,36)(25,42,32,40)(28,46)(29,44)(33,43)(37,48)(38,39)(45,47);(1,4,5,12)(2,7,8,11)(3,9,10,16)(6,13,14,15)(17,21,22,35)(18,25,26,32)(19,28,29,33)(20,30,31,34)(23,37,38,45)(24,40,41,42)(27,43,44,46)(36,47,39,48)

Polycyclic presentation:

2^10|a^2=g,b^2=fgij,c^2=hij,d^2=i,b^a=bf,c^a=cgh,c^b=ce,d^a=defi,d^b=def,d^c=dfgi,e^a=ei,e^b=ej,e^c=ei,e^d=eij,f^a=fj,f^b=fj,f^c=fj,f^d=fij,g^b=gj,g^d=gij,h^b=hi,h^d=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249477, #72249481, #72249487, …
maximal normal subgroup of	#72249477, #72249481, #72249487, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);