#7624795

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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
prime radical	#61475	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1207 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531991);