#6442376

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

1674 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)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,47)(31,42)(33,46)(34,40)(35,43)(36,44)(37,45)(38,41)(39,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,30)(22,32)(23,38)(25,42)(26,39)(27,29)(28,46)(31,36)(33,35)(37,48)(40,44)(41,43)(45,47);(10,14)(13,16)(17,20,22,36)(18,25,27,44)(19,30,32,48)(21,23,37,35)(24,39,29,47)(26,28,45,43)(31,38,34,33)(40,46,42,41);(17,21,22,37)(18,26,27,45)(19,31,32,34)(20,35,36,23)(24,40,29,42)(25,43,44,28)(30,33,48,38)(39,41,47,46);(1,3,4,7)(2,6,5,8)(18,27)(24,29)(25,44)(26,45)(28,43)(39,47)(40,42)(41,46)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=fi,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, 6349572);