Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,26)(7,17,20,31,21,30,18,13)(8,22,23,32,25,12,24,14)(19,29,35,27,34,28,33,36);(6,12)(7,18)(8,23)(9,10)(13,28)(14,26)(15,32)(16,22)(17,27)(19,34)(20,21)(24,25)(29,31)(30,36);(2,4)(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,27)(14,29)(15,30)(16,31)(17,26)(22,36)(23,33)(24,35)(25,34)(28,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250371, #72250378, #72250380, … |
| maximal normal subgroup of | #72250371, #72250378, #72250380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54898);