Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,21)(9,24)(10,13)(11,18)(12,16)(14,22)(17,32)(19,28)(20,35)(23,33)(25,36)(26,31)(27,30)(29,34);(5,7,16,30,9,20,23,32,10,19,31,36,22,34,8,18)(6,11,21,29,14,25,26,28,13,17,33,35,24,27,12,15);(5,6)(7,17,19,11)(8,12)(9,14)(10,13)(15,18,28,32)(16,21)(20,27,34,25)(22,24)(23,26)(29,30,35,36)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320338, #72320357, #72320421, … |
| maximal normal subgroup of | #72320338, #72320357, #72320421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56412);