Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,15)(10,22)(11,14)(12,18)(17,32)(19,27)(20,29)(21,28)(23,26)(24,30)(25,34)(31,35)(33,36);(5,7)(6,12)(8,17)(9,18)(10,23)(11,19)(13,25)(14,26)(15,27)(16,22)(20,33)(21,35)(24,34)(28,36)(29,32)(30,31);(5,8)(6,13)(7,17)(9,20)(10,24)(11,21)(12,25)(14,28)(15,29)(16,31)(18,33)(19,35)(22,30)(23,34)(26,36)(27,32);(6,14)(12,26)(13,28)(15,22)(16,27)(25,36)(29,30)(31,32);(5,9,11,10)(6,15,14,22)(7,18,19,23)(8,20,21,24)(12,27,26,16)(13,29,28,30)(17,33,35,34)(25,32,36,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,f^b=fi,g^b=gi,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72236660, #72237505, #72237513, … |
| maximal normal subgroup of | #72236660, #72237505, #72237513, … |
| maximal quotient of | #72236708, #72249679, #72249821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277942);