Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17198 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)(5,6,10,15,11,14,9,23)(7,16,18,12,19,27,24,26)(8,13,22,29,20,28,21,33)(17,30,32,25,31,36,34,35);(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,25)(14,26)(15,27)(16,23)(20,31)(21,34)(22,32)(28,35)(29,36)(30,33);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,20)(12,25)(14,28)(15,29)(16,30)(18,32)(19,31)(23,33)(24,34)(26,35)(27,36);(8,20)(13,28)(17,31)(21,22)(25,35)(29,33)(30,36)(32,34);(6,14)(12,26)(13,28)(15,23)(16,27)(25,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224751, #72226740, #72237258, … |
| maximal normal subgroup of | #72224751, #72226740, #72237258, … |
| maximal quotient of | #72239234, #72262590, C2×#6370728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277924);