Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,17,28,15)(7,18,21,12,23,31,22,32)(8,13,25,30,26,35,20,34)(9,27,19,14,24,29,36,33);(5,7,10,21,11,23,28,22)(6,12,15,18,17,32,16,31)(8,24,25,36,26,9,20,19)(13,33,34,29,35,14,30,27);(2,4)(5,8,10,25,11,26,28,20)(6,13,16,30,17,35,15,34)(7,19,21,24,23,36,22,9)(12,29,31,33,32,27,18,14);(5,9,10,19,11,24,28,36)(6,14,15,27,17,33,16,29)(7,20,21,8,23,25,22,26)(12,30,18,13,32,34,31,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281073);