#6354832

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354774
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1593 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:

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

Polycyclic presentation:

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

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, 6262028);