Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1325 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)(7,17)(8,22,24,28)(9,13,26,23)(10,15)(11,16)(12,20)(14,29)(18,25,35,30)(19,34,36,33)(21,31)(27,32);(5,7,11,21)(6,12,16,32)(8,18,24,35)(9,25,26,30)(10,27,29,20)(13,33,23,34)(14,31,15,17)(19,28,36,22);(2,4)(5,8,10,23,11,24,29,13)(6,9,15,28,16,26,14,22)(7,18,20,33,21,35,27,34)(12,30,31,36,32,25,17,19);(2,4)(5,9)(6,13)(7,19)(8,15)(10,28)(11,26)(12,18)(14,24)(16,23)(17,34)(20,30)(21,36)(22,29)(25,27)(31,33)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eghi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297610);