#7623510

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
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 ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1305 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)(19,23)(20,24)(21,34)(22,27)(25,31)(26,32)(28,36)(29,35)(30,44)(33,40)(37,43)(38,42)(39,45)(41,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,37)(26,38)(27,36)(31,43)(32,42)(33,41)(34,44)(39,47)(40,46)(45,48);(3,10)(6,9)(11,16)(13,15)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,43)(30,42)(33,45)(34,39)(36,46)(37,38)(41,48)(44,47);(2,7)(6,9)(8,14)(13,15)(19,28)(23,36)(29,41)(30,42)(35,46)(37,47)(38,44)(43,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|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=cf,d^b=di,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, 7530706);