#7514893

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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	#61185	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,24)(13,16)(14,19)(15,31)(17,28)(18,29)(20,30)(22,35)(23,34)(25,33)(26,36)(27,38)(32,37)(39,40);(2,5)(6,8)(9,11,14,13)(10,16,19,21)(12,25,27,40)(15,29,30,23)(17,35,37,36)(18,31,34,20)(22,28,26,32)(24,39,38,33);(1,3,4,7)(2,6,5,8)(9,12,14,27)(10,17,19,37)(11,22,13,26)(15,32,30,28)(16,33,21,39)(18,36,34,35)(20,38,31,24)(23,25,29,40);(9,13,14,11)(10,18,19,34)(12,26,27,22)(15,23,30,29)(16,20,21,31)(17,36,37,35)(24,33,38,39)(25,28,40,32);(2,5)(6,8)(10,19)(16,21)(17,37)(18,34)(20,31)(24,38)(33,39)(35,36)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,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, 7422089);