#6669696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
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	#61171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,28)(20,33)(21,38)(22,27)(23,26)(24,32)(25,37)(29,42)(30,41)(31,39)(34,45)(35,44)(36,43)(40,46)(47,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,32)(18,24,27,28)(20,34,36,46)(21,30,23,39)(25,42,44,47)(26,41,38,31)(29,37,48,35)(33,40,43,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,36)(18,25,27,44)(19,29,32,48)(21,35,23,37)(24,40,28,45)(26,43,38,33)(30,46,39,34)(31,47,41,42);(2,8)(6,9)(7,14)(13,15)(17,21,22,23)(18,26,27,38)(19,30,32,39)(20,35,36,37)(24,41,28,31)(25,43,44,33)(29,46,48,34)(40,42,45,47);(3,11)(6,9)(10,16)(13,15)(17,18)(19,31)(20,25)(21,26)(22,27)(23,38)(24,30)(28,39)(29,47)(32,41)(33,37)(34,45)(35,43)(36,44)(40,46)(42,48)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^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,g^c=gi,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, 6576892);