#8922752

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

4677 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)(3,5)(4,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18,22,24)(19,23,27,31)(20,25,26,21)(28,29,30,32);(2,7)(3,8)(10,14)(11,15)(17,19)(18,23)(20,30)(21,29)(22,27)(24,31)(25,32)(26,28);(1,3,6,8)(2,4,7,5)(9,11,13,15)(10,12,14,16)(18,24)(19,27)(21,25)(28,30);(1,4,6,5)(2,3,7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,23)(20,30)(21,29)(22,27)(24,31)(25,32)(26,28);(1,2)(3,4)(5,8)(6,7)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,29,31,32);(1,5,6,4)(2,8,7,3)(17,21,22,25)(18,26,24,20)(19,29,27,32)(23,28,31,30)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8829948);