#6757817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#37846	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6665013);