#8772568

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

4557 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679764);