Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,22,19,23,17,32)(15,20,25,36,26,37,33,35)(16,27,31,39,29,41,30,21)(24,38,28,34,40,44,42,43);(6,9)(10,12)(13,15)(14,20)(16,28)(17,33)(18,25)(19,26)(21,38)(22,36)(23,37)(24,30)(27,34)(29,42)(31,40)(32,35)(39,44)(41,43);(1,2,3,4)(13,16,17,30,19,29,18,31)(14,21,22,27,23,39,32,41)(15,24,25,28,26,40,33,42)(20,34,35,38,37,43,36,44);(5,7,8,11)(6,10,9,12)(16,29)(21,39)(24,40)(27,41)(28,42)(30,31)(34,43)(38,44)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265730);