#7545938

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	#3041	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^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,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, 7453134);