#6686591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61289	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,24)(20,34)(21,26)(22,27)(23,29)(25,37)(28,40)(30,39)(31,43)(32,36)(33,44)(35,45)(38,46)(41,47)(42,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,27)(21,31)(22,34)(23,33)(25,41)(26,43)(28,36)(29,44)(30,35)(32,40)(37,47)(38,48)(39,45)(42,46);(17,20,21,37,23,38,40,35)(18,25,28,34,29,45,26,46)(19,30,31,27,33,47,32,48)(22,36,39,44,42,43,41,24);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,40)(18,26,29,28)(19,31,33,32)(20,35,38,37)(22,39,42,41)(24,43,44,36)(25,34,45,46)(27,30,48,47);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,23,42)(18,27,29,48)(19,25,33,45)(20,36,38,43)(21,39,40,41)(24,37,44,35)(26,30,28,47)(31,34,32,46)

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593787);