#7557376

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

1651 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)(9,10,14,20)(11,16,24,33)(12,25,27,37)(13,19,29,18)(15,23,31,22)(17,34,36,28)(21,35,26,30)(32,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,11,13,22,14,24,29,23)(10,15,18,16,20,31,19,33)(12,26,28,39,27,21,34,38)(17,35,25,40,36,30,37,32);(2,7)(3,8)(12,27)(17,36)(21,26)(25,37)(28,34)(30,35)(32,40)(38,39);(1,4,5,6)(2,8,7,3)(9,11)(10,16)(12,21)(13,22)(14,24)(15,19)(17,32)(18,31)(20,33)(23,29)(25,35)(26,27)(28,38)(30,37)(34,39)(36,40);(9,12,13,28,14,27,29,34)(10,17,18,25,20,36,19,37)(11,21,22,38,24,26,23,39)(15,30,16,32,31,35,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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, 7464572);