#6448650

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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3368 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)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,30)(23,31)(24,32)(25,33)(29,35)(34,38)(36,39)(37,40);(9,11,14,21)(10,15,18,28)(12,19,23,35)(13,20,25,37)(16,29,31,26)(17,27,33,40)(22,34,24,36)(30,39,32,38);(2,5)(6,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,32)(18,31)(20,36)(21,35)(22,25)(27,39)(28,29)(30,33)(34,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,30)(18,31)(20,34)(21,35)(24,25)(27,38)(28,29)(32,33)(36,37)(39,40);(9,13,14,25)(10,17,18,33)(11,20,21,37)(12,22,23,24)(15,27,28,40)(16,30,31,32)(19,34,35,36)(26,38,29,39)

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^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, 6355846);