Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| 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×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,12,22,26)(9,17)(10,18)(11,14)(13,23)(16,21,19,28)(20,30,34,32)(24,31)(25,35,36,33)(27,29);(2,4)(5,7,15,6,11,18,10,14)(8,19,21,12,22,16,28,26)(9,17,29,13,24,31,23,27)(20,32,33,25,34,30,35,36);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,22)(13,25)(14,26)(15,28)(17,30)(18,19)(23,33)(24,34)(27,36)(29,35)(31,32);(2,4)(5,9,15,29,11,24,10,23)(6,13,18,31,14,27,7,17)(8,20,28,35,22,34,21,33)(12,25,19,32,26,36,16,30)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=efg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279966);