Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 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)(5,6,11,17,12,18,10,16)(7,19,23,13,24,33,22,34)(8,25,27,35,28,32,21,14)(9,29,30,31,26,15,20,36);(5,7)(6,13)(8,26)(9,28)(10,23)(11,22)(12,24)(14,29)(15,35)(16,33)(17,19)(18,34)(20,21)(25,36)(27,30)(31,32);(5,8,11,27,12,28,10,21)(6,14,16,32,18,35,17,25)(7,20,22,26,24,30,23,9)(13,31,33,15,34,36,19,29);(5,9,10,30,12,26,11,20)(6,15,17,31,18,29,16,36)(7,21,23,28,24,27,22,8)(13,32,19,14,34,25,33,35)Polycyclic presentation:
2^9|a^2=h,c^2=fi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292352);