#6355520

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
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
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1589 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,23,26)(19,29,31,40)(20,34,36,42)(21,28,22,27)(24,33,39,32)(25,37,41,35)(30,44,45,38)(43,47,48,46);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,19,21,32,23,31,22,33)(18,24,27,29,26,39,28,40)(20,30,35,46,36,45,37,47)(25,38,34,43,41,44,42,48);(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(18,26)(19,31)(25,41)(27,28)(30,45)(32,33)(34,42)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,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, 6262716);