Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1386 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)(7,18)(8,24)(9,28)(10,31)(11,16)(12,17)(13,21)(14,25)(15,30)(19,29)(20,26)(22,33)(23,35)(27,36)(32,34);(2,4)(5,7)(6,13)(8,15)(9,20)(10,22)(11,21)(12,23)(14,30)(16,35)(17,33)(18,31)(19,26)(24,29)(25,34)(27,28)(32,36);(5,8)(6,14)(7,19)(9,18)(10,26)(11,25)(12,27)(13,29)(15,21)(16,36)(17,20)(22,28)(23,34)(24,31)(30,35)(32,33);(5,9,10,32,12,30,11,29)(6,15,31,19,17,28,16,34)(7,20,22,36,23,14,21,24)(8,18,26,33,27,35,25,13)Polycyclic presentation:
2^9|d^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257124, #72257164, #72257167 |
| maximal normal subgroup of | #72257124, #72257164, #72257167 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321086);