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 | |
6927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,10,12)(7,14,8,11)(9,17,18,13)(15,19,16,20)(21,22)(23,28)(24,25)(26,31)(27,29)(30,36)(32,33)(34,35);(1,2)(3,4)(5,7,10,8)(6,11,12,14)(9,15,18,16)(13,19,17,20)(21,23,27,26)(22,28,29,31)(24,32,30,34)(25,33,36,35);(6,12)(7,8)(13,17)(15,16)(22,29)(24,30)(28,31)(32,34);(5,8,10,7)(6,11,12,14)(9,16,18,15)(13,19,17,20)(21,24,27,30)(22,25,29,36)(23,32,26,34)(28,33,31,35);(5,6)(7,14)(8,11)(9,13)(10,12)(15,20)(16,19)(17,18)(21,25)(22,30)(23,33)(24,29)(26,35)(27,36)(28,34)(31,32);(5,9)(6,13)(7,15)(8,16)(10,18)(11,19)(12,17)(14,20)(21,26,27,23)(22,31,29,28)(24,34,30,32)(25,35,36,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627367);