#7624221

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	#61084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

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

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, 7531417);