#7623576

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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1265 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,25)(22,36)(23,28)(27,34)(29,37)(30,38)(31,41)(32,42)(33,39)(35,40)(43,45)(44,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,22)(18,24,28,36)(20,31,32,29)(21,30,35,34)(25,38,40,27)(26,41,42,37)(33,45,46,44)(39,43,48,47);(3,10)(6,9)(11,16)(13,15)(18,25)(19,22)(21,34)(24,27)(26,39)(28,40)(29,31)(30,35)(33,45)(36,38)(37,43)(41,47)(42,48)(44,46);(2,7)(6,9)(8,14)(13,15)(17,20,23,32)(18,26,28,42)(19,29,22,31)(21,33,35,46)(24,37,36,41)(25,39,40,48)(27,43,38,47)(30,44,34,45);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,32)(26,42)(29,31)(33,46)(37,41)(39,48)(43,47)(44,45)

Polycyclic presentation:

2^9|b^2=i,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,d^b=di,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, 7530772);