#6811868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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	#60695	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,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,g^c=gi

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, 6719064);