#9938080

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

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=g,f^2=gh,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, 9845276);