Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,30)(24,26)(25,27)(28,34)(29,33)(31,36)(32,35);(5,7,9,6)(8,11,12,10)(13,15)(14,17)(16,20)(18,19)(22,26)(27,29)(28,35)(30,36);(5,8,9,12)(6,10,7,11)(15,20)(17,19)(22,27)(25,33)(26,29)(28,36)(30,35)(32,34);(6,7)(10,11)(14,18)(15,20)(21,23,24,31)(22,28,26,35)(25,34,33,32)(27,36,29,30)Polycyclic presentation:
2^9|a^2=h,b^2=egi,c^2=egi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294285);