Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4534 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)(5,6,11,16,12,7,10,15)(8,19,21,17,23,30,22,13)(9,14,26,29,27,18,25,24)(20,32,33,31,34,36,35,28);(5,7)(6,12)(8,17)(9,24)(10,16)(11,15)(13,23)(14,26)(18,25)(19,22)(20,32)(21,30)(27,29)(28,35)(31,33)(34,36);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,22)(12,23)(14,28)(15,19)(16,30)(18,31)(24,32)(25,33)(26,35)(27,34)(29,36);(5,9)(6,14)(7,18)(8,20)(10,25)(11,26)(12,27)(13,28)(15,24)(16,29)(17,31)(19,32)(21,33)(22,35)(23,34)(30,36);(5,10,12,11)(6,15,7,16)(8,21,23,22)(9,25,27,26)(13,19,17,30)(14,24,18,29)(20,33,34,35)(28,32,31,36)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,g^c=gi,h^c=hi,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224796, #72314246 |
| maximal normal subgroup of | #72224796, #72314246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286397);