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 | |
2795 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)(15,20)(16,21)(17,19)(18,22)(23,36)(24,35)(25,31)(26,38)(27,32)(28,33)(29,34)(30,37)(39,44)(40,42)(41,43);(2,4)(5,7,8,11)(6,9,10,12)(14,19)(15,23)(16,26)(18,28)(20,34)(21,37)(25,29)(27,30)(31,36)(32,38)(35,42)(39,41);(6,10)(9,12)(13,15)(14,20)(16,24)(17,25)(18,29)(19,31)(21,35)(22,34)(23,28)(26,39)(27,40)(30,41)(32,42)(33,36)(37,43)(38,44);(13,16,17,27)(14,21,19,32)(15,24,25,40)(18,30,28,26)(20,35,31,42)(22,37,33,38)(23,39,29,41)(34,43,36,44);(15,25)(20,31)(23,29)(24,40)(34,36)(35,42)(39,41)(43,44)Polycyclic presentation:
2^9|a^2=fi,b^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,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, 6347725);