#6686097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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	#60934	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,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,h^c=hi,h^e=hi

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, 6593293);