#9945540

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852736);