#7623508

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1315 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,25)(21,33)(22,37)(23,27)(24,31)(26,35)(29,44)(30,45)(32,43)(34,40)(36,47)(38,42)(39,46)(41,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(19,29)(24,39)(28,44)(30,38)(31,46)(34,48)(40,41)(42,45);(3,10)(6,9)(11,16)(13,15)(17,19)(18,24)(20,30)(21,34)(22,38)(23,29)(25,40)(26,42)(27,39)(28,33)(31,35)(32,46)(36,48)(37,41)(43,45)(44,47);(2,7)(6,9)(8,14)(13,15)(17,20,23,22)(18,25,27,37)(19,30,29,38)(21,32,36,35)(24,40,39,41)(26,33,43,47)(28,45,44,42)(31,34,46,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cfi,g^c=gh,h^c=hi

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, 7530704);