Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1412 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)(7,12)(8,10)(9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,32)(24,29)(25,35)(26,31)(27,28)(30,33)(34,36);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,23,26,27)(22,28,31,32)(24,34,25,33)(29,30,35,36);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(21,24,26,25)(22,29,31,35)(23,33,27,34)(28,36,32,30);(21,25,26,24)(22,30,31,36)(23,33,27,34)(28,35,32,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299538);