#6442238

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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

1710 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)(17,18,23,27)(19,24,31,40)(20,25,35,43)(21,36,22,26)(28,37,45,48)(29,38,30,39)(32,41,46,47)(33,44,34,42);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,31)(18,24,27,40)(20,32,35,46)(21,30,22,29)(25,41,43,47)(26,38,36,39)(28,33,45,34)(37,44,48,42);(10,14)(13,16)(17,20,23,35)(18,25,27,43)(19,28,31,45)(21,34,22,33)(24,37,40,48)(26,44,36,42)(29,32,30,46)(38,41,39,47);(19,29)(21,22)(24,38)(26,36)(28,32)(30,31)(33,34)(37,41)(39,40)(42,44)(45,46)(47,48);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,36)(19,30,31,29)(20,33,35,34)(24,38,40,39)(25,42,43,44)(28,32,45,46)(37,47,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^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,e^c=ei,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349434);