#8088188

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

4685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995384);