Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,9,12)(7,8,11,10)(13,14,19,24)(15,20,28,36)(16,21,30,38)(17,22,18,23)(25,33,42,44)(26,34,41,43)(27,35,32,40)(29,37,31,39);(5,7,9,11)(6,10,12,8)(13,15,19,28)(14,20,24,36)(16,26,30,41)(17,27,18,32)(21,34,38,43)(22,35,23,40)(25,31,42,29)(33,39,44,37);(2,4)(5,8,9,10)(6,11,12,7)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,33)(22,37)(23,39)(24,38)(26,32)(27,41)(28,42)(34,40)(35,43)(36,44);(2,4)(6,12)(7,11)(13,16)(14,21)(15,26)(17,31)(18,29)(19,30)(20,34)(22,39)(23,37)(24,38)(25,32)(27,42)(28,41)(33,40)(35,44)(36,43);(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,30,31)(20,35,36,40)(21,37,38,39)(25,41,42,26)(33,43,44,34)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409274);