Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,20)(9,24)(10,15)(11,18)(12,19)(13,22)(14,25)(17,31)(21,26)(23,34)(27,33)(28,35)(29,32)(30,36);(1,3,4,2)(5,7)(6,11,15,27)(8,17)(9,18)(10,19)(12,14,16,24)(13,26,30,31)(20,29,34,28)(21,23)(22,32)(25,33)(35,36);(5,7)(6,12,15,16)(8,21)(9,18)(10,19)(11,24,27,14)(13,29,30,28)(17,23)(20,31,34,26)(22,35)(25,33)(32,36);(5,8,9,22,10,23,25,36)(6,13,14,20,15,30,24,34)(7,17,18,32,19,21,33,35)(11,26,16,29,27,31,12,28)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265169, #72265185, #72265222, … |
| maximal normal subgroup of | #72265169, #72265185, #72265222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277222);