#6449110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3617 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)(9,10,14,18)(11,15,21,29)(12,16,24,31)(13,17,26,32)(19,27,22,34)(20,28,25,37)(23,30,36,39)(33,38,35,40);(9,11)(10,15)(12,22,24,19)(13,25,26,20)(14,21)(16,27,31,34)(17,28,32,37)(18,29)(23,33)(30,38)(35,36)(39,40);(2,5)(6,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,31)(20,33)(21,22)(25,35)(26,36)(28,38)(29,34)(32,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,20)(12,23)(14,26)(15,28)(16,30)(18,32)(19,33)(21,25)(22,35)(24,36)(27,38)(29,37)(31,39)(34,40);(12,24)(16,31)(19,22)(23,36)(27,34)(30,39)(33,35)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6356306);