#7624951

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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1177 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)(19,30)(20,35)(21,26)(22,28)(23,38)(24,27)(25,33)(29,36)(31,44)(32,42)(34,46)(37,40)(39,41)(43,48)(45,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,31)(21,39)(22,33)(23,34)(24,32)(26,42)(27,41)(28,30)(29,43)(35,48)(36,44)(37,46)(38,45)(40,47);(17,20,36,28,24,37,40,27,22,29,35,18,21,38,23,26)(19,31,44,30,32,46,47,41,33,43,48,25,39,45,34,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,24)(18,26,28,27)(19,32,33,39)(20,37,29,38)(23,35,40,36)(25,41,30,42)(31,45,43,46)(34,44,47,48);(2,8)(6,9)(7,16)(14,15)(17,21,22,24)(18,27,28,26)(19,32,33,39)(20,38,29,37)(23,35,40,36)(25,42,30,41)(31,46,43,45)(34,44,47,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,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,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, 7532147);