#8196945

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

5013 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)(3,5)(4,6)(7,8)(9,10,13,14)(11,12,15,16)(17,18,23,25)(19,26,21,24)(20,29,31,22)(27,28,30,32);(2,3)(6,7)(10,14)(11,15)(17,19,23,21)(18,24,25,26)(20,27,31,30)(22,28,29,32);(4,8)(6,7)(9,11)(10,12)(13,15)(14,16)(18,25)(22,29)(24,26)(28,32);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,16)(14,15)(17,20)(18,22)(19,27)(21,30)(23,31)(24,28)(25,29)(26,32);(17,21,23,19)(18,26,25,24)(20,30,31,27)(22,32,29,28);(1,4,5,8)(2,6,3,7)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,31)(26,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=ghi,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, 8104141);