Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,9,16,10,15)(7,12,20,28,18,29,19,27)(8,21,24,32,23,31,25,13)(17,30,34,35,33,36,22,26);(1,2)(3,4)(5,7)(6,12)(8,22)(9,18)(10,20)(11,19)(13,30)(14,27)(15,28)(16,29)(17,25)(21,26)(23,34)(24,33)(31,35)(32,36);(2,4)(5,8,9,23)(6,13)(7,17,18,33)(10,24,11,25)(12,26)(14,31)(15,21)(16,32)(19,34,20,22)(27,30)(28,36)(29,35);(6,14,16,15)(7,18)(12,27,29,28)(13,31,32,21)(17,33)(19,20)(22,34)(26,30,35,36)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=ei,f^2=i,g^2=i,h^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,g^b=gi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255710, #72256181, #72256189, … |
| maximal normal subgroup of | #72255710, #72256181, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290934);