#8533124

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

4361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,6,4)(3,7,5,8)(9,10)(11,16)(12,18)(13,23)(14,19)(15,17)(20,22)(21,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,17)(13,20)(14,21)(15,18)(19,24)(22,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,6,5)(2,8,4,7)(10,17)(11,18)(13,19)(21,22)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4,6,2)(3,7,5,8)(9,12,15,16)(10,18,17,11)(13,22,19,21)(14,24,23,20)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,5,6,3)(2,7,4,8)(9,13,15,19)(10,14,17,23)(11,20,18,24)(12,22,16,21)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,14)(10,19)(11,21)(12,20)(13,17)(15,23)(16,24)(18,22)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440320);