#6629379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#32889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3940 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)(3,6)(4,5)(7,8)(11,19)(13,24)(15,27)(18,32)(20,36)(21,25)(22,26)(28,39)(29,33)(31,34)(35,37)(38,40);(2,5)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,27)(20,28)(21,29)(22,31)(23,30)(24,32)(25,33)(26,34)(35,38)(36,39)(37,40);(9,11)(10,15)(12,20)(13,25)(14,21)(16,28)(17,29)(18,33)(19,24)(22,35)(23,37)(26,36)(27,32)(30,40)(31,38)(34,39);(9,12,14,23)(10,16,17,30)(11,20,21,37)(13,26,24,22)(15,28,29,40)(18,34,32,31)(19,35,25,36)(27,38,33,39);(1,3,4,7)(2,6,5,8)(10,17)(15,29)(16,30)(18,32)(27,33)(28,40)(31,34)(38,39)

Polycyclic presentation:

2^9|d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536575);