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×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1491 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)(5,6,11,16)(7,17,21,31)(8,22,25,13)(9,14,10,15)(12,19,28,20)(18,33,35,29)(23,27,34,36)(24,30,26,32);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,29)(14,17)(15,31)(16,28)(18,26)(22,33)(24,35)(25,34)(27,30)(32,36);(5,8,11,25)(6,13)(7,18,21,35)(9,26,10,24)(12,27)(14,30)(15,32)(16,22)(17,33)(19,34,20,23)(28,36)(29,31);(6,14)(7,19)(9,10)(13,30)(15,16)(17,31)(18,34)(20,21)(22,32)(23,35)(24,26)(29,33)Polycyclic presentation:
2^9|a^2=h,c^2=egi,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, 293117);