#8181164

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

4921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8088360);