Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,23,25,19)(16,27,28,36)(18,31,30,22)(20,32,35,29)(24,39,40,44)(26,34,42,38)(33,41,37,43);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,33)(21,23)(22,34)(27,44)(28,40)(29,41)(30,42)(31,38)(32,43)(35,37)(36,39);(2,4)(13,16)(14,20)(15,24)(17,28)(18,32)(19,33)(21,35)(22,36)(23,37)(25,40)(26,43)(27,31)(29,30)(34,39)(38,44)(41,42);(2,4)(16,28)(20,35)(24,40)(27,36)(29,32)(33,37)(39,44)(41,43);(5,7,8,11)(6,10,9,12)(15,25)(19,23)(24,40)(26,42)(33,37)(34,38)(39,44)(41,43)Polycyclic presentation:
2^9|a^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347381);