#6357813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
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

1866 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:

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

Polycyclic presentation:

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

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, 6265009);