#6448399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355595);