Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22,25,30)(9,14)(10,15)(11,16)(12,20)(13,23,28,24)(18,26,34,36)(19,27)(21,29)(31,33,35,32);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,29)(16,27)(22,31)(23,32)(24,33)(25,34)(28,36)(30,35);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,22)(15,30)(16,28)(17,31)(19,32)(20,33)(21,34)(27,36)(29,35);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,27,29)(13,22,28,30)(18,32,34,33)(26,31,36,35)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237648, #72237649, #72237672, … |
| maximal normal subgroup of | #72237648, #72237649, #72237672, … |
| maximal quotient of | #72250425, #72257727, #72257781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264689);