Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15,18,11)(8,19)(9,13)(10,14)(12,21)(16,33,34,26)(17,31,24,27)(20,36,35,28)(22,32)(23,30)(25,29);(1,3,4,2)(5,7)(6,11)(8,20)(9,18)(10,24)(12,28)(13,15)(14,31)(16,22)(17,23)(19,36)(21,35)(25,34)(26,29)(27,30)(32,33);(5,8)(6,12)(7,16)(9,21)(10,25)(11,26)(13,19)(14,32)(15,33)(17,35)(18,34)(20,24)(22,23)(27,36)(28,31)(29,30);(5,7)(6,11)(8,16)(9,18)(10,17)(12,26)(13,15)(14,27)(19,33)(20,22)(21,34)(23,24)(25,35)(28,29)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280941, #72281044, #72281219, … |
| maximal normal subgroup of | #72280941, #72281044, #72281219, … |
| maximal quotient of | #72313249 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272323);