Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,21)(16,29)(17,35)(18,23)(19,24)(20,25)(22,32)(26,43)(27,37)(28,38)(30,40)(31,36)(33,42)(34,41)(39,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,30)(17,18)(19,27)(20,28)(22,39)(23,35)(24,37)(25,38)(26,33)(29,40)(31,34)(32,44)(36,41)(42,43);(13,16,18,32,20,34,19,33)(14,22,24,29,25,42,23,41)(15,26,27,31,28,44,17,30)(21,36,35,43,38,40,37,39);(2,4)(13,17,18,15,20,27,19,28)(14,21,24,35,25,38,23,37)(16,31,32,44,34,30,33,26)(22,40,29,39,42,36,41,43);(5,8,9,7)(6,11,12,10)(13,18,20,19)(14,23,25,24)(15,17,28,27)(16,32,34,33)(21,35,38,37)(22,41,42,29)(26,30,44,31)(36,43,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470752);