Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,22)(17,23)(18,33)(19,24)(20,25)(26,35)(27,36)(28,38)(29,39)(30,40)(31,41)(32,42)(34,37)(43,44);(1,2,3,4)(13,15)(14,21)(16,27)(17,32)(18,28)(19,34)(20,29)(22,36)(23,42)(24,37)(25,39)(26,30)(31,43)(33,38)(35,40)(41,44);(5,7,8,12)(6,9,10,11)(13,16,19,30,20,31,18,17)(14,22,24,40,25,41,33,23)(15,26,28,27,29,32,34,43)(21,35,38,36,39,42,37,44);(13,17,18,31,20,30,19,16)(14,23,33,41,25,40,24,22)(15,27,34,26,29,43,28,32)(21,36,37,35,39,44,38,42);(13,18,20,19)(14,24,25,33)(15,28,29,34)(16,17,31,30)(21,37,39,38)(22,40,41,23)(26,27,32,43)(35,44,42,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373437);