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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(13,14,19,25)(15,27,31,21)(16,32,35,41)(17,36,33,43)(18,26,20,24)(22,34,42,29)(23,37,40,28)(30,39,38,44);(2,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,33)(17,35)(18,30)(19,31)(20,38)(22,40)(23,42)(24,39)(25,27)(26,44)(28,29)(32,36)(34,37)(41,43);(13,16,18,34,19,35,20,29)(14,22,24,41,25,42,26,32)(15,28,38,17,31,37,30,33)(21,36,44,23,27,43,39,40);(1,2,3,4)(13,17,20,28,19,33,18,37)(14,23,26,36,25,40,24,43)(15,29,30,35,31,34,38,16)(21,32,39,42,27,41,44,22);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,24,25,26)(15,30,31,38)(16,34,35,29)(17,28,33,37)(21,39,27,44)(22,41,42,32)(23,36,40,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=h,e^2=f,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=cg,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, 7521397);