#7624404

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,34)(22,38)(23,28)(25,30)(27,36)(29,39)(31,42)(32,41)(33,45)(35,40)(37,48)(43,47)(44,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,32)(21,30)(22,29)(23,31)(25,34)(26,41)(27,40)(28,42)(33,47)(35,36)(37,46)(38,39)(43,45)(44,48);(17,19)(18,25)(20,29)(21,35)(22,32)(23,31)(24,34)(26,40)(27,41)(28,44)(30,36)(33,46)(37,47)(38,43)(39,45)(42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,20,23,22)(18,26,28,38)(19,29,31,32)(21,33,37,36)(24,39,42,41)(25,40,44,43)(27,34,45,48)(30,35,46,47)

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,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531600);