Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1494 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)(7,12,9,14)(8,20,22,32)(10,16)(11,15)(13,23,30,25)(17,33)(18,26)(19,28,24,31)(21,36)(27,35)(29,34);(1,2)(3,4)(5,7,10,24,11,9,18,19)(6,12,26,28,15,14,16,31)(8,17,25,34,22,21,23,35)(13,27,20,36,30,29,32,33);(2,4)(5,8)(6,13)(7,17)(9,21)(10,25)(11,22)(12,27)(14,29)(15,30)(16,32)(18,23)(19,35)(20,26)(24,34)(28,36)(31,33);(5,9,10,19,11,7,18,24)(6,14,26,31,15,12,16,28)(8,21,25,35,22,17,23,34)(13,29,20,33,30,27,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268680);