#8105343

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

4458 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)(3,6)(4,7)(5,8)(9,10,15,17)(11,18,21,16)(12,13,22,14)(19,23,24,20)(25,26)(27,30)(28,32)(29,31)(33,36)(34,37)(35,38)(39,40);(1,3,6,2)(4,7,5,8)(9,11,15,21)(10,16,17,18)(12,19,22,24)(13,20,14,23)(25,27)(26,30)(28,34)(29,33)(31,36)(32,37)(35,39)(38,40);(1,4,6,5)(2,7,3,8)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,23)(21,24)(26,31)(30,36)(32,38)(37,40);(1,5,6,4)(2,7,3,8)(9,13,15,14)(10,12,17,22)(11,20,21,23)(16,19,18,24)(27,33)(30,36)(34,39)(37,40);(9,14,15,13)(10,12,17,22)(11,20,21,23)(16,24,18,19)(26,31)(30,36)(32,38)(37,40);(25,28,29,35)(26,32,31,38)(27,34,33,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=gi,f^2=h,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, 8012539);