#6448312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 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)(9,10,13,15)(11,14,12,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,44)(45,47)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,31)(21,27)(22,30)(24,40)(25,38)(26,36)(28,32)(29,34)(33,46)(35,45)(37,42)(39,43)(41,48)(44,47);(9,12,13,11)(10,14,15,16)(19,27)(21,34)(23,36)(26,43)(28,31)(29,30)(32,35)(37,40)(38,39)(42,44)(45,46)(47,48);(17,20,22,33)(18,24,25,41)(19,28,30,45)(21,35,34,32)(23,37,38,47)(26,44,43,42)(27,31,29,46)(36,40,39,48);(1,3,4,7)(2,6,5,8)(18,25)(23,38)(24,41)(26,43)(36,39)(37,47)(40,48)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355508);