#7557378

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	#61266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=g,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,e^b=ei,h^b=hi,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, 7464574);