#8195965

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

4903 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)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,29,31)(20,21,26,25)(27,30,32,28);(1,2,5,4)(3,6,8,7)(9,10,12,14)(11,13,16,15)(18,23)(20,26)(24,31)(27,32);(2,4)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,27)(21,30)(22,29)(23,31)(25,28)(26,32);(9,11,12,16)(10,13,14,15)(17,20,22,26)(18,25,23,21)(19,27,29,32)(24,28,31,30);(1,3,5,8)(2,6,4,7);(1,4,5,2)(3,7,8,6)(17,21,22,25)(18,26,23,20)(19,28,29,30)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^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, 8103161);