#7500738

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

2711 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,26)(13,28)(14,31)(15,20)(16,21)(18,29)(19,30)(22,23)(24,36)(25,37)(27,33)(32,35)(34,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,27)(13,23)(14,24)(15,32)(16,25)(18,38)(19,34)(20,35)(21,37)(22,28)(26,33)(29,39)(30,40)(31,36);(9,12)(10,18)(11,22)(13,21)(14,28)(15,29)(16,19)(17,33)(20,30)(23,36)(24,34)(25,39)(26,31)(27,32)(35,38)(37,40);(2,6)(3,8)(9,13)(10,19)(11,23)(12,21)(14,26)(15,30)(16,18)(17,34)(20,29)(22,36)(24,33)(25,38)(27,37)(28,31)(32,40)(35,39);(9,14,16,15)(10,20,21,31)(11,24,25,32)(12,28,19,29)(13,26,18,30)(17,35,37,36)(22,34,39,27)(23,33,38,40)

Polycyclic presentation:

2^9|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, 7407934);