#7501143

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=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, 7408339);