Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
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 | |
1883 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)(7,18)(8,22,24,26)(9,25,29,15)(10,30)(11,16)(12,17)(13,19)(14,23,33,31)(20,32)(21,36)(27,34,28,35);(1,2)(3,4)(5,7,10,19,12,21,11,20)(6,13,16,18,17,32,30,36)(8,15,23,34,24,25,31,35)(9,26,27,33,29,22,28,14);(2,4)(5,8)(6,14)(7,15)(9,13)(10,23)(11,31)(12,24)(16,26)(17,33)(18,27)(19,34)(20,35)(21,25)(22,30)(28,36)(29,32);(5,9)(6,15)(7,14)(8,13)(10,28)(11,27)(12,29)(16,35)(17,25)(18,31)(19,22)(20,26)(21,33)(23,36)(24,32)(30,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290418);