#8195898

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

4761 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)(3,8)(4,6)(5,7)(17,18,23,22)(19,24,29,28)(20,30,25,21)(26,32,31,27);(1,3,5,4)(2,6,7,8)(9,10,13,12)(11,15,16,14)(17,19,23,29)(18,24,22,28)(20,26,25,31)(21,27,30,32);(2,7)(6,8)(10,12)(14,15)(17,20)(18,21)(19,26)(22,30)(23,25)(24,27)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32);(1,4,5,3)(2,8,7,6)(17,21,23,30)(18,25,22,20)(19,27,29,32)(24,31,28,26);(1,2)(3,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,22,23,18)(19,28,29,24)(20,30,25,21)(26,32,31,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=h,f^2=ghi,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, 8103094);