Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,23,25,28,34,30)(24,31,33,36,32,29,35,27);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,26,28,25,30,34,22)(24,27,33,31,32,36,35,29);(1,2,3,4)(7,11)(10,12)(14,18)(17,20)(21,24)(22,27)(23,29)(25,32)(26,35)(28,36)(30,31)(33,34);(5,7,8,11)(6,10,9,12)(24,32)(27,36)(29,31)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290123);