#7623436

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623323
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
prime radical	#60718	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1248 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,28)(19,25,31,45)(20,26,22,38)(21,34,37,48)(24,39,42,30)(27,33,46,36)(29,43,32,44)(35,40,47,41);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,24)(20,29)(21,35)(22,32)(23,31)(25,34)(26,40)(27,44)(28,42)(30,36)(33,39)(37,47)(38,41)(43,46)(45,48);(2,8)(6,9)(7,16)(14,15)(17,19)(18,25)(20,32)(21,30)(22,29)(23,31)(24,34)(26,44)(27,40)(28,45)(33,47)(35,36)(37,39)(38,43)(41,46)(42,48);(17,20,23,22)(18,26,28,38)(19,29,31,32)(21,33,37,36)(24,40,42,41)(25,43,45,44)(27,34,46,48)(30,35,39,47)

Polycyclic presentation:

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

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, 7530632);