#7030391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6937587);