#6445893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,24,32,40)(20,25,35,43)(21,26,22,27)(29,37,31,39)(30,38,45,47)(33,41,46,48)(34,42,36,44);(2,5)(6,8)(17,19,23,32)(18,24,28,40)(20,33,35,46)(21,31,22,29)(25,41,43,48)(26,39,27,37)(30,36,45,34)(38,44,47,42);(9,11,12,16)(10,13,14,15)(19,29)(21,22)(24,37)(26,27)(30,33)(31,32)(34,36)(38,41)(39,40)(42,44)(45,46)(47,48);(1,3,4,7)(2,6,5,8)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,38)(26,42)(27,44)(28,43)(29,33)(31,46)(32,45)(37,41)(39,48)(40,47);(17,21,23,22)(18,26,28,27)(19,29,32,31)(20,34,35,36)(24,37,40,39)(25,42,43,44)(30,33,45,46)(38,41,47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,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, 6353089);