Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137659 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
756 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)(5,6,8,13,21,31,35,29,10,14,22,30,9,15,20,12)(7,11,17,26,33,36,32,23,19,25,16,24,18,27,34,28);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,19)(12,28)(13,26)(14,25)(15,27)(16,22)(20,34)(21,33)(23,29)(24,30)(31,36)(32,35);(6,12)(8,20)(9,21)(11,23)(13,15)(14,29)(16,32)(17,34)(18,33)(22,35)(24,27)(25,28)(26,36)(30,31)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,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240719, #72240725, #72240727, … |
| maximal normal subgroup of | #72240719, #72240725, #72240727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55387);