#7985991

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893187);