#7624332

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1256 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)(19,33)(20,36)(21,28)(22,32)(23,30)(24,41)(25,29)(26,34)(27,37)(31,43)(35,45)(38,44)(39,47)(40,46)(42,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,34)(18,26,30,33)(20,22,21,25)(24,35,40,42)(27,31,44,47)(28,32,36,29)(37,39,38,43)(41,48,46,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,21)(18,27,30,44)(19,22,34,25)(24,42,40,35)(26,31,33,47)(28,41,36,46)(29,45,32,48)(37,39,38,43);(3,11)(6,9)(10,16)(13,15)(17,21,23,20)(18,28,30,36)(19,22,34,25)(24,38,40,37)(26,29,33,32)(27,41,44,46)(31,45,47,48)(35,39,42,43);(2,8)(6,9)(7,14)(13,15)(17,22,23,25)(18,29,30,32)(19,20,34,21)(24,39,40,43)(26,36,33,28)(27,45,44,48)(31,46,47,41)(35,37,42,38)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,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,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531528);