#6357433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357427
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264629);