#8099336

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4615 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8006532);