#7563593

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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
prime radical	C₂×AGL(1,ℤ/8)	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=gi,e^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,h^d=hi

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, 7470789);