Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,23)(10,14)(11,17)(12,21)(13,25)(16,30)(18,29)(20,34)(22,28)(24,27)(26,31)(32,35)(33,36);(5,7)(6,11)(8,20)(9,24)(10,18)(12,26)(13,29)(14,27)(15,23)(16,21)(17,25)(19,32)(22,31)(28,34)(30,33)(35,36);(5,8,9,22,10,21,25,36)(6,12,13,28,14,19,23,33)(7,16,17,31,18,20,24,35)(11,26,15,30,27,32,29,34);(1,3,4,2)(8,21)(12,19)(16,20)(22,36)(26,32)(28,33)(30,34)(31,35)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266299);