Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,11,16,9,12)(7,13,19,29,20,27,17,25)(8,21,23,30,24,26,22,14)(18,32,33,36,34,35,31,28);(6,12,16,15)(7,17,20,19)(13,27)(14,26,30,21)(18,31,34,33)(25,29)(28,36)(32,35);(1,2,3,4)(5,7)(6,13)(8,18)(9,17)(10,19)(11,20)(12,25)(14,28)(15,29)(16,27)(21,32)(22,31)(23,33)(24,34)(26,35)(30,36);(5,8,9,22,11,24,10,23)(6,14,15,21,16,30,12,26)(7,18,17,31,20,34,19,33)(13,28,29,32,27,36,25,35)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269452);