Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,18)(8,22,25,14)(9,13)(10,26)(11,16)(12,17)(15,21)(19,30,34,33)(20,29)(23,36,35,28)(24,31,27,32);(1,2,4,3)(5,7,10,9,12,21,11,20)(6,13,16,18,17,29,26,15)(8,23,27,19,25,35,24,34)(14,30,32,36,22,33,31,28);(5,8)(6,14)(7,19)(9,23)(10,24)(11,27)(12,25)(13,28)(15,30)(16,31)(17,22)(18,33)(20,35)(21,34)(26,32)(29,36);(1,3,4,2)(5,9,10,21,12,20,11,7)(6,15,16,13,17,18,26,29)(8,23,24,34,25,35,27,19)(14,30,31,28,22,33,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,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, 272246);