#6817486

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814819
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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,d^2=f,e^2=fhi,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,e^c=ei

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, 6724682);