#7489896

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 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)(3,6)(4,5)(7,8)(17,18,23,25,22,28,21,27)(19,24,34,39,29,38,31,36)(20,26,35,40,32,42,33,41)(30,37,45,47,43,48,44,46);(2,5)(3,7)(10,14)(11,12)(17,19)(18,24)(20,30)(21,31)(22,29)(23,34)(25,39)(26,37)(27,36)(28,38)(32,43)(33,44)(35,45)(40,47)(41,46)(42,48);(9,10)(11,16)(12,15)(13,14)(18,25)(19,29)(20,32)(21,23)(24,36)(26,41)(27,28)(37,47)(38,39)(40,42)(44,45)(46,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,32)(18,26,28,42)(19,30,29,43)(21,33,23,35)(24,37,38,48)(25,40,27,41)(31,44,34,45)(36,46,39,47);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(19,29)(20,32)(24,38)(26,42)(31,34)(33,35)(36,39)(40,41)

Polycyclic presentation:

2^9|a^2=gi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7397092);