#8101973

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

4610 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,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, 8009169);