#7623385

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1678 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)(11,21)(12,19)(13,16)(14,28)(15,31)(17,24)(18,23)(20,30)(22,38)(25,37)(26,33)(27,34)(29,35)(32,39)(36,40);(1,3,4,8)(2,5,6,7)(10,16)(17,32)(18,21)(19,31)(20,33)(28,34)(35,36)(37,38);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,29)(15,25)(16,32)(18,34)(19,35)(20,37)(21,28)(24,30)(26,39)(27,40)(31,36)(33,38);(2,6)(5,7)(9,11)(10,18)(12,25)(13,23)(14,24)(15,22)(16,21)(17,34)(19,38)(20,36)(26,40)(27,39)(28,32)(29,30)(31,37)(33,35);(9,12,13,15)(10,19,16,31)(11,22,23,25)(14,26,27,30)(17,35,32,36)(18,37,21,38)(20,28,33,34)(24,29,39,40)

Polycyclic presentation:

2^9|b^2=fi,e^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=cfi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530581);