#8195805

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

4879 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=gh,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^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103001);