#7504047

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
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

1764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,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, 7411243);