Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,15,22,33,35,26,10,14,24,13,9,16,23,30)(7,11,18,27,25,34,21,31,20,28,17,32,19,29,36,12);(1,2,3,4)(5,7)(6,11)(8,18)(9,19)(10,20)(12,30)(13,32)(14,28)(15,27)(16,29)(17,24)(21,35)(22,25)(23,36)(26,31)(33,34);(5,7)(6,12)(8,21)(9,25)(10,20)(11,26)(13,29)(14,31)(15,34)(16,27)(17,23)(18,35)(19,22)(24,36)(28,30)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240725, #72240727, #72240732, … |
| maximal normal subgroup of | #72240725, #72240727, #72240732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55391);