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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
966 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)(7,16)(8,21)(9,25)(10,15)(11,14)(12,19)(13,23)(17,27)(18,29)(20,28)(22,30)(24,32)(26,31)(33,34)(35,36);(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,29)(15,28)(16,25)(17,23)(21,36)(24,34)(26,35)(27,32)(31,33);(5,8,9,23,11,24,10,26)(6,13,15,21,14,31,25,32)(7,17,18,34,20,35,19,22)(12,27,28,30,29,36,16,33);(6,14)(7,18,20,19)(12,16,29,28)(13,31)(15,25)(17,34,35,22)(21,32)(27,33,36,30)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292622);