Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1366 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)(5,6,11,16)(7,17,14,9)(8,20)(10,26,28,15)(12,27,30,19)(13,25)(18,34)(21,36)(22,31)(23,32)(24,33)(29,35);(2,4)(5,7)(6,12)(8,21)(9,15)(10,27)(11,14)(13,22)(16,30)(17,26)(18,24)(19,28)(20,34)(23,31)(25,35)(29,33)(32,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,25)(12,29)(14,31)(15,32)(16,20)(17,34)(19,35)(21,27)(24,28)(26,33)(30,36);(5,9,10,12,11,17,28,30)(6,14,26,19,16,7,15,27)(8,22,23,29,25,34,24,36)(13,31,33,35,20,18,32,21)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239720, #72252628, #72256869, … |
| maximal normal subgroup of | #72239720, #72252628, #72256869, … |
| maximal quotient of | #72239730 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265885);