#6448653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437632
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	C₂⁴	C₂⁵
prime radical	#62856	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3293 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)(9,10)(11,15)(12,16)(13,17)(14,18)(19,27)(20,28)(21,29)(22,31)(23,32)(24,33)(25,34)(26,35)(30,37)(36,39)(38,40);(9,11,14,21)(10,15,18,29)(12,19,23,37)(13,24,26,20)(16,30,32,27)(17,33,35,28)(22,38,25,36)(31,39,34,40);(2,5)(6,8)(9,12)(10,16)(11,19)(13,25)(14,23)(15,27)(17,34)(18,32)(20,38)(21,37)(22,26)(24,36)(28,40)(29,30)(31,35)(33,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,27)(17,31)(18,32)(20,36)(21,37)(24,38)(25,26)(28,39)(29,30)(33,40)(34,35);(9,13,14,26)(10,17,18,35)(11,20,21,24)(12,22,23,25)(15,28,29,33)(16,31,32,34)(19,36,37,38)(27,39,30,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355849);