#6673803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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	#23956	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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)(19,25)(20,33)(21,27)(22,28)(23,40)(24,29)(26,35)(30,42)(31,43)(32,44)(34,41)(36,47)(37,46)(38,48)(39,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,34)(21,30)(22,36)(23,31)(24,32)(26,45)(27,42)(28,47)(29,44)(33,41)(35,39)(37,38)(40,43)(46,48);(17,20,24,37)(18,26,29,47)(19,22,32,39)(21,35,23,36)(25,41,44,48)(27,46,40,33)(28,42,45,43)(30,34,31,38);(2,7)(6,9)(8,15)(14,16)(17,21,24,23)(18,27,29,40)(19,30,32,31)(20,35,37,36)(22,34,39,38)(25,42,44,43)(26,46,47,33)(28,41,45,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,24,39)(18,28,29,45)(19,20,32,37)(21,38,23,34)(25,33,44,46)(26,42,47,43)(27,48,40,41)(30,36,31,35)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=fi,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,e^d=ei,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, 6580999);