#6801628

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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)(3,7)(4,6)(5,8)(17,18,23,35,22,26,21,27)(19,28,34,24,31,41,30,38)(20,25,36,46,32,39,33,40)(29,42,45,37,43,48,44,47);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(20,32)(25,39)(29,43)(33,36)(37,47)(40,46)(42,48)(44,45);(10,13)(14,15)(17,19,22,31)(18,24,26,38)(20,29,32,43)(21,34,23,30)(25,37,39,47)(27,41,35,28)(33,45,36,44)(40,48,46,42);(2,6)(5,7)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,37)(26,39)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(38,47)(41,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(18,26)(20,32)(24,38)(27,35)(28,41)(29,43)(33,36)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708824);