Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2430 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,24,27,25,28,32,35)(23,29,31,36,30,26,33,34);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(23,30)(26,29)(31,33)(34,36);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,30)(26,29)(31,33)(34,36);(7,15)(10,13)(14,20)(17,19);(6,12)(10,13)(11,18)(17,19)(21,23,24,31,25,30,32,33)(22,26,27,34,28,29,35,36)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=df,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72308036, #72308060, #72308094, … |
| maximal normal subgroup of | #72308036, #72308060, #72308094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306281);