#9930256

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

4401 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)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,33)(28,39)(29,40)(30,34)(31,32)(35,36)(37,38);(1,2,4,6)(3,5,8,7)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,27)(26,33)(28,29)(30,36)(31,37)(32,38)(34,35)(39,40);(10,16)(12,17)(15,22)(18,23)(25,28,32,30)(26,34,31,39)(27,36,38,29)(33,40,37,35);(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,29,32,36)(26,35,31,40)(27,28,38,30)(33,34,37,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,30,32,28)(26,34,31,39)(27,36,38,29)(33,35,37,40);(1,3,4,8)(2,5,6,7)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,31,32,26)(27,37,38,33)(28,39,30,34)(29,40,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9837452);