#8701240

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

4819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608436);