#7611880

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

1925 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,18)(13,19)(15,21)(16,22)(17,24)(20,32)(23,27)(25,31)(26,33)(28,36)(29,37)(30,34)(35,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,15,24,16,25,32,19)(10,17,20,14,22,13,21,31)(12,23,28,33,29,38,40,34)(18,26,35,27,37,30,36,39);(2,6)(3,7)(9,12,16,29)(10,18,22,37)(11,23,25,38)(13,30,17,26)(14,27,31,39)(15,28,32,40)(19,34,24,33)(20,35,21,36);(2,6)(5,8)(9,13,15,31,16,17,32,14)(10,19,21,25,22,24,20,11)(12,26,28,27,29,30,40,39)(18,33,36,23,37,34,35,38);(9,14,15,13,16,31,32,17)(10,19,20,11,22,24,21,25)(12,27,28,30,29,39,40,26)(18,34,35,23,37,33,36,38)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=h,e^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei

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, 7519076);