Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,15,24,11)(7,12,19,25,18,28,17,27)(8,20,23,13,22,30,21,29)(16,31,32,35,33,36,34,26);(1,2)(3,4)(6,11)(8,21)(9,24)(12,25)(13,29)(14,15)(16,32)(17,19)(22,23)(26,35)(27,28)(33,34);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,25)(13,26)(14,27)(15,28)(19,24)(20,31)(21,32)(22,33)(23,34)(29,35)(30,36);(5,8,10,22)(6,13,15,29)(7,16,18,33)(9,23,24,21)(11,20,14,30)(12,26,28,35)(17,34,19,32)(25,31,27,36)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290321);