#9947470

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

4375 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)(9,10,14,17)(11,15,19,12)(13,21,23,16)(18,20,24,22)(25,26,32,34)(27,28,29,30)(31,39,40,33)(35,38,37,36);(1,2,4,6)(3,5,8,7)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,32)(30,34)(31,35)(33,36)(37,40)(38,39);(9,11,14,19)(10,12,17,15)(13,18,23,24)(16,20,21,22)(25,28,32,30)(26,27,34,29)(31,36,40,38)(33,35,39,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,21)(23,24)(25,29,32,27)(26,28,34,30)(31,37,40,35)(33,36,39,38);(1,2)(3,7)(4,6)(5,8)(9,12,14,15)(10,11,17,19)(13,22,23,20)(16,24,21,18)(25,30,32,28)(26,27,34,29)(31,36,40,38)(33,37,39,35);(1,3,4,8)(2,5,6,7)(9,13,14,23)(10,16,17,21)(11,18,19,24)(12,20,15,22)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,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, 9854666);