Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,9,12,10,14,8,13)(7,15,16,20,17,19,18,11);(7,16,17,18)(11,19,20,15);(5,7)(6,11,14,20)(8,16)(9,18)(10,17)(12,19,13,15);(6,12)(8,9)(11,19)(13,14)(15,20)(16,18)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286523, #72286985, #72287040, … |
| maximal normal subgroup of | #72286523, #72286985, #72287040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293607);