Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,18,10,17)(7,19,23,26,24,33,22,13)(8,25,27,14,29,21,28,34)(9,15,31,35,32,36,30,20);(5,7,10,22,12,24,11,23)(6,13,16,19,18,26,17,33)(8,15,27,35,29,36,28,20)(9,14,30,25,32,34,31,21);(1,3,4,2)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,32)(15,22)(16,25)(17,21)(18,34)(19,31)(23,36)(24,35)(30,33);(5,9,12,32)(6,15,18,36)(7,21,24,25)(8,26,29,13)(10,31,11,30)(14,22,34,23)(16,20,17,35)(19,28,33,27)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296324);