#8195963

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

4973 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,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, 8103159);