#7623313

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
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	#60458	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1622 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)(11,19)(12,22)(13,24)(15,27)(17,32)(18,31)(20,37)(21,36)(23,26)(25,35)(28,39)(29,40)(30,38)(33,34);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,27)(20,29)(21,28)(22,32)(23,33)(24,31)(25,30)(26,34)(35,38)(36,39)(37,40);(3,8)(5,7)(9,11,14,21)(10,15,16,28)(12,23,25,37)(13,26,24,20)(17,33,30,40)(18,34,31,29)(19,35,36,22)(27,38,39,32);(2,6)(5,7);(10,16)(15,28)(17,30)(18,31)(27,39)(29,34)(32,38)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530509);