#6448668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437478
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	#62855	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318726, #72318970
maximal normal subgroup of	#72318726, #72318970

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355864);