#9947383

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

4583 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,26)(27,30)(28,31)(29,34)(32,38)(33,36)(35,40)(37,39);(1,2,4,6)(3,5,8,7)(10,14)(15,22)(16,19)(23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,38)(35,39)(37,40);(10,14)(15,22)(16,19)(23,24)(26,27)(31,33)(32,34)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,3,4,8)(2,5,6,7)(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)

Polycyclic presentation:

2^9|b^2=ghi,d^2=i,e^2=h,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, 9854579);