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 | |
6733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(7,15)(9,17)(12,16)(13,18)(21,22,26,29)(23,27,30,34)(24,25,28,33)(31,32,35,36);(1,2,3,4)(5,6,11,8)(7,16,15,12)(9,13,17,18)(10,14,20,19)(23,30)(27,34)(31,35)(32,36);(5,7,11,15)(6,12,8,16)(9,10,17,20)(13,19,18,14)(21,23)(22,27)(24,31)(25,32)(26,30)(28,35)(29,34)(33,36);(5,8,11,6)(7,12,15,16)(9,18,17,13)(10,19,20,14);(5,9)(6,13)(7,10)(8,18)(11,17)(12,19)(14,16)(15,20)(21,24)(22,25)(23,31)(26,28)(27,32)(29,33)(30,35)(34,36);(5,10)(6,14)(7,17)(8,19)(9,15)(11,20)(12,13)(16,18)(21,25)(22,28)(23,32)(24,29)(26,33)(27,35)(30,36)(31,34)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759931);