#6686125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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	#3054	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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)(17,18,22,25)(19,29,30,24)(20,33,32,44)(21,26,37,41)(23,28,34,27)(31,46,47,40)(35,42,36,48)(38,45,39,43);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,30)(18,24,25,29)(20,34,32,23)(21,36,37,35)(26,42,41,48)(27,33,28,44)(31,39,47,38)(40,43,46,45);(17,20,34,19,22,32,23,30)(18,24,28,44,25,29,27,33)(21,35,39,31,37,36,38,47)(26,40,45,48,41,46,43,42);(2,7)(6,9)(8,15)(14,16)(18,25)(19,30)(20,32)(26,41)(27,28)(31,47)(35,36)(43,45);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,22,37)(18,26,25,41)(19,31,30,47)(20,35,32,36)(23,38,34,39)(24,40,29,46)(27,43,28,45)(33,42,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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,d^b=di,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, 6593321);