Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3102 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)(5,6,11,16,12,18,10,17)(7,19,20,13,22,30,21,31)(8,14,29,32,25,34,24,33)(9,26,27,15,28,36,23,35);(5,7)(6,13)(8,23)(9,24)(10,20)(11,21)(12,22)(14,26)(15,33)(16,19)(17,30)(18,31)(25,27)(28,29)(32,35)(34,36);(2,4)(5,8)(6,14)(7,9)(10,24)(11,29)(12,25)(13,15)(16,32)(17,33)(18,34)(19,26)(20,27)(21,23)(22,28)(30,36)(31,35);(2,4)(5,9)(6,15)(7,8)(10,27)(11,23)(12,28)(13,14)(16,26)(17,36)(18,35)(19,32)(20,24)(21,29)(22,25)(30,33)(31,34);(5,10,12,11)(6,16,18,17)(7,20,22,21)(8,24,25,29)(9,27,28,23)(13,19,31,30)(14,32,34,33)(15,26,35,36)Polycyclic presentation:
2^9|a^2=fi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,h^b=hi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408109);