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 | |
6394 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)(7,15)(8,10)(9,12)(11,14)(13,20)(16,17)(18,19)(21,22)(23,28)(24,26)(25,32)(27,30)(29,33)(31,34)(35,36);(1,2,3,4)(5,7,11,9)(6,12,14,15)(8,18,13,16)(10,17,20,19)(23,30)(27,28)(31,36)(34,35);(7,9)(12,15)(16,18)(17,19)(21,23,26,30)(22,27,24,28)(25,31,33,36)(29,34,32,35);(5,8,11,13)(6,10,14,20)(7,16,9,18)(12,19,15,17)(21,23)(22,28)(24,27)(25,31)(26,30)(29,35)(32,34)(33,36);(5,9,11,7)(6,12,14,15)(8,18,13,16)(10,19,20,17)(21,24,26,22)(23,27,30,28)(25,32,33,29)(31,34,36,35);(5,10)(6,13)(7,17)(8,14)(9,19)(11,20)(12,16)(15,18)(21,25)(22,29)(23,31)(24,32)(26,33)(27,34)(28,35)(30,36)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622104);