#7625125

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=i,e^2=f,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532321);