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 | |
7492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,10,12)(7,13,14,8)(9,11,18,20)(15,19,17,16)(21,22,26,27)(23,29,30,25)(24,32,34,28)(31,33,36,35);(1,2)(3,4)(5,7)(6,8)(9,17)(10,14)(11,19)(12,13)(15,18)(16,20)(21,23,26,30)(22,25,27,29)(24,31,34,36)(28,33,32,35);(7,14)(8,13)(15,17)(16,19)(22,27)(23,30)(28,32)(31,36);(5,7)(6,8)(9,15)(10,14)(11,16)(12,13)(17,18)(19,20)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,35)(30,36);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,22)(23,25)(24,28)(26,27)(29,30)(31,33)(32,34)(35,36);(5,9)(6,11)(7,15)(8,16)(10,18)(12,20)(13,19)(14,17)(21,25)(22,23)(24,33)(26,29)(27,30)(28,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72336917 |
| maximal normal subgroup of | #72336917 |
| maximal quotient of | #72337000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646694);