#72249463

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

3795 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=gh,c^2=fgh,d^2=j,b^a=bg,c^a=ce,c^b=cf,d^a=dfgij,d^b=defhj,d^c=defh,e^b=ei,e^d=ej,f^b=fij,f^c=fij,f^d=fij,g^c=gi,g^d=gij,h^c=hj,h^d=hi

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.9);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);