Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1607 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)(8,22)(9,27)(10,14)(11,15)(12,19)(13,26)(17,21)(18,30)(20,29)(23,28)(24,31)(25,35)(32,34)(33,36);(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,17)(15,29)(16,27)(18,24)(22,32)(25,34)(26,33)(28,31)(35,36);(2,4)(5,6)(7,17)(8,13)(9,14)(10,27)(11,15)(12,19)(16,21)(18,32)(20,29)(22,26)(23,28)(24,31)(25,35)(30,34)(33,36);(5,8,9,24,11,26,10,25)(6,13,14,31,15,22,27,35)(7,18,19,23,21,34,20,33)(12,28,16,30,29,36,17,32)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327073);