#8131212

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

4404 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)(3,4)(5,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,26)(27,35)(28,29)(30,32)(31,36)(33,34)(37,40)(38,39);(2,6)(3,7)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,36)(27,32,30,35)(28,29,34,33)(37,38,39,40);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,30)(26,32,36,35)(28,39,34,37)(29,40,33,38);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,31,34)(26,33,36,29)(27,37,30,39)(32,40,35,38);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,29,31,33)(26,34,36,28)(27,38,30,40)(32,39,35,37);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,30,31,27)(26,35,36,32)(28,39,34,37)(29,40,33,38)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8038408);