#6802637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24218	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709833);