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 | |
6332 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)(5,6)(7,12)(8,13)(9,14)(10,11)(15,19)(16,17)(18,20)(21,22,27,26)(23,30,25,28)(24,29,33,34)(31,36,32,35);(1,2,3,4)(21,23,27,25)(22,28,26,30)(24,32,33,31)(29,36,34,35);(6,11)(9,19)(12,17)(13,20)(23,25)(28,30)(31,32)(35,36);(5,7)(6,12)(8,14)(9,13)(10,16)(11,17)(15,18)(19,20)(21,24)(22,29)(23,31)(25,32)(26,34)(27,33)(28,35)(30,36);(5,8)(6,13)(7,15)(9,17)(10,18)(11,20)(12,19)(14,16)(21,25,27,23)(22,30,26,28)(24,31,33,32)(29,35,34,36);(5,9,10,19)(6,14,11,15)(7,13,16,20)(8,17,18,12)(21,26,27,22)(23,30,25,28)(24,29,33,34)(31,35,32,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998952);