#7624008

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

1358 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)(11,16)(12,22)(13,26)(14,30)(15,19)(17,24)(18,28)(20,36)(21,35)(23,34)(25,38)(27,31)(29,40)(32,39)(33,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,19,30)(12,23,25,20)(13,21,29,28)(17,32,33,31)(18,26,35,40)(22,34,38,36)(24,39,37,27);(9,12)(10,17)(11,20)(13,27)(14,23)(15,25)(16,31)(18,34)(19,33)(21,37)(22,26)(24,28)(29,39)(30,32)(35,36)(38,40);(2,5)(6,8)(9,11,15,14)(10,16,19,30)(12,20,25,23)(13,21,29,28)(17,31,33,32)(18,26,35,40)(22,36,38,34)(24,27,37,39);(1,3,4,7)(2,6,5,8)(9,11,15,14)(10,16,19,30)(12,20,25,23)(13,21,29,28)(17,31,33,32)(18,26,35,40)(22,36,38,34)(24,27,37,39)

Polycyclic presentation:

2^9|b^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	#72260370, #72260372, #72260392, …
maximal normal subgroup of	#72260370, #72260372, #72260392, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531204);