#7624603

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#62440	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531799);