#7545930

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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	#3508	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1673 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)(11,22)(12,17)(13,28)(14,20)(15,21)(16,24)(18,29)(19,26)(23,34)(25,32)(27,31)(30,35)(33,40)(36,38)(37,39);(1,3,5,2)(4,6,8,7)(9,11,14,24,15,12,26,25)(10,16,19,22,21,32,20,17)(13,27,29,40,30,23,38,39)(18,34,28,33,36,31,35,37);(1,4,5,8)(2,6,3,7);(2,3)(6,7)(9,12)(10,17)(11,15)(13,27)(14,25)(16,19)(18,33)(20,32)(21,22)(23,30)(24,26)(28,31)(29,40)(34,35)(36,37)(38,39);(9,13,14,29,15,30,26,38)(10,18,19,28,21,36,20,35)(11,23,24,39,12,27,25,40)(16,31,22,37,32,34,17,33)

Polycyclic presentation:

2^9|a^2=g,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,e^b=ei,h^b=hi,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, 7453126);