#8100329

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

4587 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^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, 8007525);