#7502558

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(11,22)(12,25)(13,17)(14,20)(15,32)(16,21)(18,29)(19,24)(23,35)(26,38)(27,40)(28,37)(30,36)(31,34)(33,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,26)(13,14)(15,24)(16,31)(18,35)(19,21)(20,22)(23,30)(25,33)(27,29)(28,39)(32,34)(36,40)(37,38);(9,12,14,28)(10,18,20,36)(11,23,13,27)(15,30,16,29)(17,33,22,38)(19,35,34,40)(21,37,32,25)(24,39,31,26);(2,7)(3,8)(9,13,14,11)(10,19,20,34)(12,27,28,23)(15,24,16,31)(17,32,22,21)(18,35,36,40)(25,38,37,33)(26,30,39,29);(1,4,5,6)(2,3,7,8)(12,28)(18,36)(23,27)(25,37)(26,39)(29,30)(33,38)(35,40)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=fi,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,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, 7409754);