#7624007

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1331 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)(6,8)(9,10,15,19)(11,20,21,37)(12,17,14,29)(13,25,28,39)(16,31,32,23)(18,24,36,27)(22,38,30,35)(26,33,40,34);(1,2)(3,6)(4,5)(7,8)(11,21)(16,32)(20,37)(22,30)(23,31)(26,40)(33,34)(35,38);(9,11)(10,16)(12,22)(13,26)(14,30)(15,21)(17,33)(18,35)(19,32)(20,25)(23,27)(24,31)(28,40)(29,34)(36,38)(37,39);(2,5)(6,8)(9,12,15,14)(10,17,19,29)(11,22,21,30)(13,24,28,27)(16,33,32,34)(18,25,36,39)(20,38,37,35)(23,26,31,40);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,17,19,29)(11,22,21,30)(13,24,28,27)(16,33,32,34)(18,25,36,39)(20,38,37,35)(23,26,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310218, #72315883, #72315916, …
maximal normal subgroup of	#72310218, #72315883, #72315916, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531203);