#7623177

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
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
prime radical	#60647	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1692 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)(3,6)(4,5)(7,8)(10,17)(11,23)(13,30)(14,31)(15,18)(19,32)(20,21)(22,25)(24,37)(27,40)(28,39)(29,33)(34,38)(35,36);(2,5)(6,8)(9,10,14,20,15,21,13,17,16,22,30,11,18,23,31,25)(12,19,28,35,29,36,27,32,26,34,40,24,33,37,39,38);(1,3,4,7)(2,6,5,8)(9,11,13,10,18,17,14,23,16,20,31,22,15,25,30,21)(12,24,27,19,33,32,28,37,26,35,39,34,29,38,40,36);(9,12)(10,19)(11,24)(13,27)(14,28)(15,29)(16,26)(17,32)(18,33)(20,35)(21,36)(22,34)(23,37)(25,38)(30,40)(31,39);(12,26)(19,34)(24,35)(27,39)(28,40)(29,33)(32,38)(36,37)

Polycyclic presentation:

2^9|b^2=g,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530373);