Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 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)(7,9)(8,21,24,31)(10,16)(11,17)(12,19)(13,23,32,33)(14,25)(15,27)(18,29,35,28)(20,26)(22,30,36,34);(5,7,10,19,11,20,27,14)(6,12,15,9,17,25,16,26)(8,18,33,30,24,35,23,34)(13,29,21,22,32,28,31,36);(1,2,3,4)(5,8,11,24)(6,13)(7,18)(9,22,26,36)(10,23,27,33)(12,28,25,29)(14,30)(15,31)(16,21)(17,32)(19,34)(20,35);(5,9,10,25,11,26,27,12)(6,14,15,7,17,19,16,20)(8,22,23,29,24,36,33,28)(13,30,31,18,32,34,21,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272972);