#7624547

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1380 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319166, #72320187, #72320683
maximal normal subgroup of	#72319166, #72320187, #72320683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531743);