Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,28)(25,34)(26,38)(27,39)(29,35)(30,36)(31,37)(33,41)(40,44)(42,43);(5,6)(7,11)(8,10)(9,12)(13,15,18,28,19,27,16,25)(14,20,21,24,23,34,32,39)(17,26,30,41,31,42,29,40)(22,33,35,38,37,44,36,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,18)(14,21,23,32)(15,25,27,28)(17,30,31,29)(20,24,34,39)(22,36,37,35)(26,41,42,40)(33,43,44,38);(7,12)(9,11)(13,17)(14,22)(15,26)(16,29)(18,30)(19,31)(20,33)(21,35)(23,37)(24,38)(25,40)(27,42)(28,41)(32,36)(34,44)(39,43);(6,10)(9,11)(15,27)(17,31)(20,34)(22,37)(24,39)(25,28)(29,30)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410105);