#7514897

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

1901 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)(11,22)(12,25)(13,29)(14,21)(15,20)(16,19)(17,24)(18,27)(23,33)(26,32)(28,36)(30,35)(31,34)(37,39)(38,40);(2,5)(6,8)(9,11)(10,17)(12,26)(13,14)(15,24)(16,30)(18,33)(19,29)(20,22)(21,35)(23,31)(25,37)(27,38)(28,39)(32,36)(34,40);(1,3,4,7)(2,6,5,8)(9,12,15,28)(10,18,20,34)(11,23,24,38)(13,26,30,39)(14,27,16,31)(17,32,22,37)(19,25,21,36)(29,33,35,40);(9,13,15,30)(10,17,20,22)(11,14,24,16)(12,26,28,39)(18,32,34,37)(19,29,21,35)(23,27,38,31)(25,33,36,40);(2,5)(6,8)(9,14,15,16)(10,19,20,21)(11,13,24,30)(12,27,28,31)(17,29,22,35)(18,25,34,36)(23,26,38,39)(32,33,37,40)

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^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, 7422093);