#7625087

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624802
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

1282 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,27)(19,23,30,26)(20,31,33,47)(21,34,35,37)(24,38,40,32)(25,41,42,29)(28,43,45,46)(36,48,39,44);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,30)(18,23,27,26)(20,28,33,45)(21,29,35,41)(24,36,40,39)(25,37,42,34)(31,43,47,46)(32,44,38,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,29,44,19,28,35,38,22,33,41,48,30,45,21,32)(18,24,34,46,26,39,42,47,27,40,37,43,23,36,25,31);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532283);