#6684165

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	#3045	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=i,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, 6591361);