#8665368

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8572564);