#7516544

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1947 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)(11,15)(12,23)(13,17)(14,19)(16,26)(18,29)(20,34)(21,31)(22,33)(24,30)(25,36)(27,32)(28,35)(37,40)(38,39);(2,5)(6,8)(9,11,14,22)(10,15,19,33)(12,24,25,38)(13,27,29,21)(16,34,35,40)(17,32,18,31)(20,28,37,26)(23,30,36,39);(9,12,14,25)(10,16,19,35)(11,20,22,37)(13,28,29,26)(15,30,33,39)(17,23,18,36)(21,38,27,24)(31,34,32,40);(1,3,4,7)(2,6,5,8)(10,17)(13,29)(15,31)(16,23)(18,19)(21,27)(24,38)(26,28)(30,34)(32,33)(35,36)(39,40);(2,5)(6,8)(12,25)(16,35)(20,37)(23,36)(24,38)(26,28)(30,39)(34,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,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, 7423740);