#6808762

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	#3061	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=f,e^2=hi,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, 6715958);