#8193866

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

4952 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)(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,20,25,24)(21,26,29,30)(27,28,32,31);(1,3,6,4)(2,7,5,8)(10,14)(11,15)(18,23)(19,25)(26,30)(27,32);(2,5)(7,8)(9,11)(10,12)(13,15)(14,16)(21,29)(26,30)(27,32)(28,31);(1,3,6,4)(2,7,5,8)(9,12)(10,11)(13,16)(14,15)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28);(1,4,6,3)(2,8,5,7)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27);(1,5,6,2)(3,8,4,7)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=gi,f^2=h,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, 8101062);