Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,10,12)(7,13,8,11)(9,14,18,17)(15,20,16,19)(24,32)(28,35)(31,34)(33,36);(1,2)(3,4)(5,7,10,8)(6,11,12,13)(9,16,18,15)(14,20,17,19)(21,22,27,30)(23,29,26,25)(24,28,32,35)(31,36,34,33);(6,12)(7,8)(14,17)(15,16)(21,23,27,26)(22,25,30,29)(24,31,32,34)(28,33,35,36);(5,8,10,7)(6,13,12,11)(9,16,18,15)(14,20,17,19)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,36)(30,35);(5,6)(7,11)(8,13)(9,17)(10,12)(14,18)(15,20)(16,19)(21,25)(22,23)(24,33)(26,30)(27,29)(28,31)(32,36)(34,35);(5,9,10,18)(6,14,12,17)(7,15,8,16)(11,19,13,20)(21,26,27,23)(22,29,30,25)(24,34,32,31)(28,36,35,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893309);