Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| 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 | |
1468 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)(7,15)(8,21,25,32)(9,26)(10,27)(11,16)(12,17)(13,19)(14,24,33,28)(18,29,35,36)(20,34)(22,30,23,31);(2,4)(5,7,10,19,12,20,11,9)(6,13,16,15,17,26,27,34)(8,22,28,18,25,23,24,35)(14,30,21,36,33,31,32,29);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,28)(12,25)(13,29)(15,31)(16,32)(17,33)(19,22)(20,35)(21,27)(26,36)(30,34);(2,4)(5,9,11,20,12,19,10,7)(6,15,27,13,17,34,16,26)(8,23,28,35,25,22,24,18)(14,31,21,29,33,30,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279959);