Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | #61030 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2526 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)(5,6,12,16,11,14,10,27)(7,17,20,13,21,32,28,30)(8,15,23,9,24,26,29,25)(18,34,36,31,35,33,22,19);(5,7)(6,13)(8,22)(9,19)(10,28)(11,21)(12,20)(14,30)(15,33)(16,32)(17,27)(18,23)(24,36)(25,31)(26,34)(29,35);(2,4)(6,14)(9,25)(13,30)(15,26)(16,27)(17,32)(19,31)(33,34);(5,8)(6,15)(7,18)(9,16)(10,29)(11,24)(12,23)(13,31)(14,26)(17,34)(19,30)(20,36)(21,35)(22,28)(25,27)(32,33);(5,9,12,26,11,25,10,15)(6,8,16,23,14,24,27,29)(7,19,20,34,21,31,28,33)(13,22,32,18,30,36,17,35)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,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320787 |
| maximal normal subgroup of | #72320787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291213);