Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| 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 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,11,17,27)(8,19,20,31)(9,15)(10,14)(12,21,29,24)(13,22)(16,32)(18,26,23,30)(25,35)(28,34)(33,36);(2,4)(5,7)(6,11)(8,16)(9,17)(10,23)(12,28)(13,26)(14,30)(15,27)(18,22)(19,32)(20,33)(21,35)(24,34)(25,29)(31,36);(1,2,3,4)(5,8)(6,12)(7,16)(9,20)(10,24)(11,25)(13,19)(14,31)(15,29)(17,33)(18,35)(21,22)(23,34)(26,36)(27,28)(30,32);(6,13)(7,17)(10,22)(11,26)(12,19)(14,15)(16,33)(21,24)(25,36)(27,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=h,c^2=egi,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293097);