#6687802

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	#61412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 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)(3,11,13,7)(4,8,5,9)(12,16,14,15)(17,18,22,26)(19,30,33,24)(20,36,38,48)(21,27,42,46)(23,29,37,28)(25,41,32,40)(31,44,47,43)(34,45,35,39);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,24)(20,27)(21,36)(22,33)(23,34)(25,44)(26,30)(28,45)(29,39)(31,41)(32,43)(35,37)(38,46)(40,47)(42,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,37,41,22,38,23,40)(18,25,29,48,26,32,28,36)(19,31,35,46,33,47,34,27)(21,39,44,30,42,45,43,24);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(18,26)(20,38)(24,30)(27,46)(28,29)(31,47)(39,45)(40,41);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,21,22,42)(18,27,26,46)(19,32,33,25)(20,39,38,45)(23,43,37,44)(24,41,30,40)(28,47,29,31)(34,48,35,36)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,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, 6594998);