#7623442

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
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	#60717	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1226 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)(19,24)(20,26)(21,34)(22,38)(23,28)(25,32)(27,36)(29,43)(30,40)(31,45)(33,44)(35,39)(37,48)(41,46)(42,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,22)(18,24,28,38)(20,30,33,29)(21,31,37,36)(25,41,42,39)(26,40,44,43)(27,34,45,48)(32,46,47,35);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,25)(19,29)(21,35)(22,30)(23,33)(24,39)(26,34)(27,40)(28,42)(31,47)(32,36)(37,46)(38,41)(43,45)(44,48);(2,8)(6,9)(7,16)(14,15)(17,20)(18,26)(19,30)(21,32)(22,29)(23,33)(24,40)(25,34)(27,39)(28,44)(31,46)(35,36)(37,47)(38,43)(41,45)(42,48);(17,19,23,22)(18,24,28,38)(20,29,33,30)(21,31,37,36)(25,39,42,41)(26,43,44,40)(27,34,45,48)(32,35,47,46)

Polycyclic presentation:

2^9|b^2=fi,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=cfi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530638);