#6854988

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(19,25)(21,29)(23,31)(24,28)(26,32)(27,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,18)(19,26)(20,22)(21,24)(23,27)(25,32)(28,29)(30,31);(9,10,13,12)(11,15,16,14)(17,19)(18,23)(20,26)(21,30)(22,27)(24,32)(25,29)(28,31);(2,6)(5,7)(17,20,22,18)(19,26,27,23)(21,28,29,24)(25,32,30,31);(3,8)(5,7)(9,12,13,10)(11,14,16,15)(18,20)(19,27)(24,28)(25,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6762184);