#6351206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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	#62925	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258402);