Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | C22×C4×D4 |
| prime radical | D4×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2540 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)(5,6,11,16,12,17,10,28)(7,18,21,13,22,31,23,32)(8,14,24,30,20,35,29,34)(9,25,19,36,27,15,26,33);(5,7)(6,13)(8,9)(10,23)(11,21)(12,22)(14,33)(15,34)(16,31)(17,32)(18,28)(19,24)(20,27)(25,30)(26,29)(35,36);(2,4);(5,8)(6,14)(7,19)(9,23)(10,29)(11,24)(12,20)(13,15)(16,30)(17,35)(18,36)(21,27)(22,26)(25,32)(28,34)(31,33);(5,9,11,19,12,27,10,26)(6,15,16,33,17,25,28,36)(7,20,21,29,22,8,23,24)(13,30,31,35,32,34,18,14)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72250686, #72250867, #72250872, … |
| maximal normal subgroup of | #72250686, #72250867, #72250872, … |
| maximal quotient of | #72318337, #72318344, C2×#6384013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291209);