#7501345

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408541);