#9947005

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

4530 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)(9,10,14,15)(11,16,19,23)(12,17,22,21)(13,20,24,18)(25,26,31,32)(27,30,28,35)(29,33,38,40)(34,39,36,37);(1,2,4,6)(3,5,8,7)(10,15)(11,19)(17,21)(18,20)(25,27)(26,30)(28,31)(29,34)(32,35)(33,39)(36,38)(37,40);(10,15)(11,19)(17,21)(18,20)(26,32)(27,28)(33,40)(34,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,23)(17,24)(18,22)(25,28,31,27)(26,30,32,35)(29,36,38,34)(33,39,40,37);(1,2)(3,7)(4,6)(5,8)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,29)(26,33)(27,34)(28,36)(30,39)(31,38)(32,40)(35,37);(1,3,4,8)(2,5,6,7)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(25,30)(26,27)(28,32)(29,37)(31,35)(33,36)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=i,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, 9854201);