#8723004

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

4886 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8630200);