Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22,19,13)(9,25,27,32)(10,28)(11,16)(12,20)(14,18,33,35)(15,29)(21,34)(23,30)(24,31,26,36);(1,2)(3,4)(5,7,11,21)(6,12,16,30)(8,18,19,35)(9,26,27,24)(10,23,29,20)(13,25,22,32)(14,31,33,36)(15,34,28,17);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,19)(12,25)(14,17)(15,31)(16,22)(20,27)(21,35)(26,29)(28,36)(30,32)(33,34);(5,9,10,18,11,27,29,35)(6,14,15,32,16,33,28,25)(7,19,20,26,21,8,23,24)(12,13,17,31,30,22,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290741);