#6350082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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	#63013	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2773 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=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, 6257278);