#7625104

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

1271 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)(19,28)(20,24)(21,35)(22,27)(23,32)(25,33)(26,34)(29,42)(30,41)(31,43)(36,38)(37,46)(39,48)(40,47)(44,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(19,29)(23,37)(28,42)(31,45)(32,46)(39,40)(43,44)(47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,30,47,20,31,21,32,22,29,33,48,34,45,36,46)(18,23,38,43,26,40,25,28,27,37,35,44,24,39,41,42);(3,12)(7,10)(11,13)(15,16)(17,20,22,34)(18,24,27,26)(19,31,29,45)(21,33,36,30)(23,39,37,40)(25,38,41,35)(28,43,42,44)(32,48,46,47);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)

Polycyclic presentation:

2^9|c^2=fghi,d^2=i,e^2=f,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, 7532300);