Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×C22⋊C4 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,17,12,18,11,16)(7,19,21,13,22,33,25,34)(8,23,26,14,27,32,30,35)(9,28,29,15,24,36,20,31);(2,4)(5,7,12,22)(6,13)(8,24,27,9)(10,21,11,25)(14,28)(15,32)(16,19)(17,33)(18,34)(20,30,29,26)(23,31)(35,36);(2,4)(5,8)(6,14)(7,20)(9,21)(10,26)(11,30)(12,27)(13,31)(15,34)(16,23)(17,32)(18,35)(19,36)(22,29)(24,25)(28,33);(2,4)(5,9,11,20,12,24,10,29)(6,15,17,36,18,31,16,28)(7,8,25,30,22,27,21,26)(13,32,33,35,34,23,19,14)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279974);