#7624971

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1184 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)(18,24)(19,28)(20,23)(22,35)(25,38)(27,39)(29,34)(30,46)(31,45)(33,44)(36,41)(37,43)(40,48)(42,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,21,24)(19,25,31,36)(20,32,23,26)(22,27,34,37)(28,41,45,38)(29,39,35,43)(30,42,33,40)(44,47,46,48);(17,19,29,46,23,30,22,28,21,31,35,44,20,33,34,45)(18,25,39,48,26,42,27,41,24,36,43,47,32,40,37,38);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(25,36)(26,32)(27,37)(38,41)(39,43)(40,42)(47,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,23)(18,26,24,32)(19,30,31,33)(22,29,34,35)(25,40,36,42)(27,43,37,39)(28,44,45,46)(38,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532167);