#72249458

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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)(3,7,11,9)(4,6)(5,8)(10,15,16,14)(12,13)(17,18)(19,27)(20,24)(21,33,36,25)(22,26)(23,30)(28,37)(29,38,34,42)(31,40,35,43)(32,41)(39,45)(44,47,46,48);(1,3)(2,7,4,9)(5,10,13,16)(6,11)(8,15)(12,14)(17,19)(18,23,26,27)(20,28,32,45)(21,34,36,29)(22,30)(24,39)(25,42)(31,46)(33,38)(35,44)(37,41)(40,47,43,48);(1,4,6,2)(3,9)(5,12)(7,11)(8,13)(10,14,16,15)(17,20,22,32)(18,24,26,41)(19,28)(21,35,36,31)(23,37)(25,43,33,40)(27,39)(29,44)(30,45)(34,46)(38,47)(42,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,36)(18,25,26,33)(19,29,30,34)(20,31,32,35)(23,38,27,42)(24,40,41,43)(28,44,45,46)(37,47,39,48)

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=hij,d^2=i,b^a=bf,c^a=cgh,c^b=cehj,d^a=defghj,d^b=dfj,d^c=dfi,e^a=ei,e^c=ei,e^d=ei,f^a=fj,f^b=fi,f^c=fj,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	#72249467, #72249481, #72249487, …
maximal normal subgroup of	#72249467, #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.5);;
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*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*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);