#6686588

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(17,18,23,22)(19,29,33,25)(20,35,39,44)(21,28,24,27)(26,40,38,37)(30,46,48,43)(31,42,34,32)(36,41,47,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,25)(20,36)(21,31)(22,29)(23,33)(24,34)(26,43)(27,42)(28,32)(30,37)(35,45)(38,46)(39,47)(40,48)(41,44);(17,20,24,37,23,39,21,40)(18,26,28,35,22,38,27,44)(19,30,34,47,33,48,31,36)(25,41,32,43,29,45,42,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,24)(18,27,22,28)(19,31,33,34)(20,37,39,40)(25,42,29,32)(26,35,38,44)(30,47,48,36)(41,43,45,46);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,23,18)(19,32,33,42)(20,38,39,26)(21,28,24,27)(25,34,29,31)(30,43,48,46)(35,37,44,40)(36,41,47,45)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,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, 6593784);