#6537403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,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, 6444599);