Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,18,11,17,10,16)(7,19,23,13,24,35,22,36)(8,25,27,14,29,21,28,31)(9,30,32,20,33,15,34,26);(1,2)(3,4)(5,7)(6,13)(8,26)(9,31)(10,23)(11,24)(12,22)(14,33)(15,27)(16,35)(17,36)(18,19)(20,29)(21,32)(25,34)(28,30);(2,4)(5,8)(6,14)(7,20)(9,13)(10,27)(11,29)(12,28)(15,22)(16,21)(17,31)(18,25)(19,32)(23,30)(24,26)(33,36)(34,35);(5,9)(6,15)(7,21)(8,13)(10,34)(11,33)(12,32)(14,22)(16,20)(17,30)(18,26)(19,28)(23,31)(24,25)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290328);