#7563592

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

1604 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)(11,17,22,16)(12,24,23,33)(13,27,30,39)(14,31,25,19)(18,35,36,29)(21,37,28,40)(26,32,38,34);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,14)(13,28)(15,22)(17,20)(18,32)(19,24)(21,30)(23,25)(26,35)(27,37)(29,38)(31,33)(34,36)(39,40);(9,12,25,22,15,23,14,11)(10,17,31,33,20,16,19,24)(13,26,35,28,30,38,29,21)(18,34,39,40,36,32,27,37);(9,13,14,29,15,30,25,35)(10,18,19,27,20,36,31,39)(11,21,23,38,22,28,12,26)(16,32,33,40,17,34,24,37);(1,4,5,3)(2,7,8,6)(9,14,15,25)(10,19,20,31)(11,12,22,23)(13,29,30,35)(16,24,17,33)(18,27,36,39)(21,26,28,38)(32,37,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=g,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, 7470788);