Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4398 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)(7,17)(8,22,25,13)(9,14)(10,15)(11,16)(12,20)(18,29,35,33)(19,28)(21,32)(23,36,34,27)(24,31,26,30);(2,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,29)(14,17)(15,32)(16,28)(18,24)(22,33)(25,34)(26,35)(27,30)(31,36);(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,30)(15,31)(16,22)(17,33)(19,34)(20,23)(21,35)(28,36)(29,32);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,25,26)(12,17,28,32)(13,30,22,31)(18,34,35,23)(27,33,36,29)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237171, #72237174, #72237194, … |
| maximal normal subgroup of | #72237171, #72237174, #72237194, … |
| maximal quotient of | #72238999, #72249715, #72267254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264782);