#6351513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6258709);