#7625106

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624812
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 ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1258 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)(19,28)(20,24)(21,34)(22,27)(23,32)(25,30)(26,33)(29,41)(31,42)(35,39)(36,46)(37,47)(38,48)(40,44)(43,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(19,29)(23,36)(28,41)(31,45)(32,46)(37,38)(42,43)(47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,30,47,20,31,35,46,22,29,44,48,33,45,21,32)(18,23,34,43,26,38,40,41,27,36,39,42,24,37,25,28);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,20,22,33)(18,24,27,26)(19,31,29,45)(21,30,35,44)(23,37,36,38)(25,39,40,34)(28,42,41,43)(32,47,46,48)

Polycyclic presentation:

2^9|c^2=fgh,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532302);