#6874120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6781316);