#7623228

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1444 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)(9,10)(11,20)(12,17)(13,27)(14,29)(15,19)(16,23)(18,21)(22,33)(24,39)(25,35)(26,32)(28,40)(30,37)(31,38)(34,36);(2,5)(6,8);(9,11,21,36,14,24,28,38,15,25,26,37,12,22,13,23)(10,16,27,33,17,30,32,35,19,31,40,39,29,34,18,20);(1,3,4,7)(2,6,5,8);(9,12,15,14)(10,17,19,29)(11,22,25,24)(13,26,28,21)(16,30,31,34)(18,27,32,40)(20,33,35,39)(23,37,38,36)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234361, #72234362, #72234365, …
maximal normal subgroup of	#72234361, #72234362, #72234365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530424);