Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| 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×#3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15)(7,16)(8,20)(9,11,23,14)(12,19)(13,27)(17,31,33,35)(18,26)(21,29)(22,28)(24,34)(25,36,30,32);(2,4)(6,11)(8,21)(9,23)(12,24)(14,15)(16,26)(17,32)(18,34)(20,28)(22,27)(31,35)(33,36);(1,2,3,4)(5,7,10,19)(6,12,15,16)(8,17,22,33)(9,18,23,34)(11,24,14,26)(13,25,29,30)(20,35,28,31)(21,32,27,36);(5,8,9,21,10,22,23,27)(6,13,14,28,15,29,11,20)(7,17,18,32,19,33,34,36)(12,25,26,31,16,30,24,35)Polycyclic presentation:
2^9|a^2=h,c^2=gi,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, 293208);