Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470368 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2818 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,24,27,34)(16,21,28,36)(17,22,18,32)(20,31,35,26)(25,38,39,44)(29,37,30,42)(33,41,43,40);(5,7)(6,9)(8,11)(10,12)(13,15,17,31,19,27,18,26)(14,20,32,34,23,35,22,24)(16,25,29,41,28,39,30,40)(21,33,42,44,36,43,37,38);(2,4)(7,11)(9,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,28)(20,33)(22,37)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(34,44)(35,43);(7,11)(9,12)(16,28)(21,36)(25,39)(29,30)(33,43)(37,42)(38,44)(40,41);(6,10)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,31)(16,29,28,30)(20,34,35,24)(21,37,36,42)(25,40,39,41)(33,44,43,38)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377568);