Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,29)(17,23)(18,34)(19,25)(20,24)(22,26)(27,40)(28,39)(30,43)(31,42)(32,44)(33,37)(35,38)(36,41);(2,4)(5,7,8,11)(6,9,10,12)(13,15,19,28)(14,21,25,39)(16,30,26,36)(17,31,33,32)(18,27,20,35)(22,41,29,43)(23,42,37,44)(24,38,34,40);(6,10)(9,12)(13,16,19,26)(14,22,25,29)(15,17,28,33)(18,32,20,31)(21,37,39,23)(24,44,34,42)(27,36,35,30)(38,41,40,43);(13,17)(14,23)(15,26)(16,28)(18,30)(19,33)(20,36)(21,22)(24,41)(25,37)(27,32)(29,39)(31,35)(34,43)(38,42)(40,44);(13,18,19,20)(14,24,25,34)(15,27,28,35)(16,31,26,32)(17,30,33,36)(21,38,39,40)(22,42,29,44)(23,41,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347736);