#8699289

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

4668 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=h,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, 8606485);