Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188759 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,15,9,25)(7,16,20,27,18,31,19,12)(8,21,22,13,24,28,23,29)(17,32,34,30,33,26,35,36);(1,2)(3,4)(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,29,32)(14,16,25,31)(21,30,28,36)(22,34)(23,35)(24,33);(2,4)(5,8)(6,13,15,29)(7,17)(9,22)(10,23)(11,24)(12,26,27,32)(14,21,25,28)(16,30,31,36)(18,33)(19,34)(20,35);(7,18)(12,27)(16,31)(17,33)(19,20)(26,32)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265741);