#6615837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(3,7)(4,8)(5,6)(17,18,22,26,23,27,21,35)(19,28,30,39,31,38,36,24)(20,25,34,41,32,40,33,45)(29,42,44,47,43,48,46,37);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,23,31)(18,24,27,39)(20,29,32,43)(21,30,22,36)(25,37,40,47)(26,38,35,28)(33,44,34,46)(41,48,45,42);(2,6)(7,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,37)(26,41)(27,40)(28,42)(30,44)(31,43)(35,45)(36,46)(38,48)(39,47);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,32)(25,40)(29,43)(33,34)(37,47)(41,45)(42,48)(44,46);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,32,34)(24,38,39,28)(25,41,40,45)(29,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6523033);