Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,11,18,12,16)(7,19,9,13,21,15,22,28)(8,14,26,30,25,32,24,31)(20,33,35,36,34,29,23,27);(1,2)(3,4)(5,7,12,22,11,21,10,9)(6,13,17,15,18,28,16,19)(8,23,24,34,25,35,26,20)(14,29,30,27,32,33,31,36);(2,4)(5,8)(6,14)(7,20)(9,23)(10,24)(11,25)(12,26)(13,27)(15,29)(16,30)(17,31)(18,32)(19,33)(21,34)(22,35)(28,36);(5,9,10,21,11,22,12,7)(6,15,16,13,18,19,17,28)(8,23,24,34,25,35,26,20)(14,29,30,27,32,33,31,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,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, 272444);