Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2324 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,33)(25,35)(26,34)(27,36)(28,38)(29,37)(30,40)(31,39)(41,43)(42,44);(2,4)(13,15)(14,19)(16,27)(17,24)(18,25)(20,36)(21,35)(22,33)(23,28)(26,31)(29,42)(30,41)(32,38)(34,39)(37,44)(40,43);(6,9)(10,12)(13,16,18,29)(14,20,21,37)(15,23,25,41)(17,30,31,28)(19,32,35,43)(22,40,39,38)(24,27,26,42)(33,36,34,44);(5,7,8,11)(6,10,9,12)(15,24)(17,31)(19,33)(22,39)(23,27)(25,26)(28,30)(32,36)(34,35)(38,40)(41,42)(43,44);(14,21)(15,25)(20,37)(22,39)(23,41)(24,26)(27,42)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6356105);