#6686590

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	#61290	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)(19,30)(20,35)(21,27)(22,26)(23,29)(24,32)(25,38)(28,44)(31,42)(33,45)(34,37)(36,40)(39,46)(41,47)(43,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,27)(21,40)(22,32)(23,33)(25,42)(26,30)(28,45)(29,37)(31,39)(34,44)(35,41)(36,47)(38,48)(43,46);(17,20,23,38,22,36,44,39)(18,25,28,35,26,46,29,40)(19,31,33,27,32,48,34,47)(21,37,42,24,41,45,43,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,26)(20,36)(24,30)(27,47)(28,29)(31,48)(37,45)(38,39);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,22,41)(18,27,26,47)(19,25,32,46)(20,37,36,45)(23,42,44,43)(24,39,30,38)(28,48,29,31)(33,35,34,40)

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=f,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, 6593786);