Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,18)(8,23)(9,27)(10,30)(11,16)(12,17)(13,20)(14,19)(15,28)(21,36)(22,32)(24,31)(25,34)(26,35)(29,33);(5,7)(6,13)(8,24)(9,26)(10,22)(11,20)(12,21)(14,33)(15,35)(16,36)(17,32)(18,30)(19,28)(23,27)(25,29)(31,34);(5,8)(6,14,17,23)(7,9)(10,26)(11,25)(12,19)(13,31,32,15)(16,35,30,34)(18,27,36,33)(20,24)(21,29)(22,28);(5,9,12,29)(6,15,17,31)(7,19,21,8)(10,24,11,28)(13,23,32,14)(16,33,30,27)(18,34,36,35)(20,25,22,26)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293116);