#7623875

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=i,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	#72321618
maximal normal subgroup of	#72321618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531071);