#6635565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	#23929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6542761);