#6442383

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	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

1672 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,36)(29,44)(30,40)(31,41)(32,42)(34,48)(35,45)(37,43)(38,46)(39,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,29)(22,31)(23,38)(25,40)(26,44)(27,41)(28,46)(30,36)(32,35)(33,48)(37,47)(39,43)(42,45);(10,14)(13,16)(17,20,22,36)(18,25,27,33)(19,29,31,47)(21,23,37,35)(24,39,41,44)(26,45,43,28)(30,38,34,32)(40,42,48,46);(17,21,22,37)(18,26,27,43)(19,30,31,34)(20,35,36,23)(24,40,41,48)(25,28,33,45)(29,32,47,38)(39,46,44,42);(1,3,4,7)(2,6,5,8)(18,27)(24,41)(25,33)(26,43)(28,45)(39,44)(40,48)(42,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,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,d^c=di,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, 6349579);