Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,19)(9,11)(10,23)(12,17)(13,21)(14,26)(16,32)(18,28)(20,25)(22,31)(24,30)(27,35)(29,34)(33,36);(5,7,10,24,11,17,26,18)(6,12,14,30,9,15,23,28)(8,16,21,35,22,33,25,34)(13,29,19,36,20,27,31,32);(5,8)(6,13)(7,16,17,33)(9,20)(10,25)(11,22)(12,27,15,29)(14,31)(18,35,24,34)(19,23)(21,26)(28,32,30,36);(2,4)(5,9,11,6)(7,15,17,12)(8,20,22,13)(10,23,26,14)(16,27,33,29)(18,30,24,28)(19,21,31,25)(32,34,36,35)Polycyclic presentation:
2^9|b^2=ef,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363147);