Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2064 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)(5,6,11,10)(7,8,9,12)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,26)(25,40)(27,35)(28,42)(29,38)(30,39)(31,44)(32,36)(34,41)(37,43);(5,7,11,9)(6,12,10,8)(13,15,18,26,19,27,17,32)(14,20,22,24,23,36,33,35)(16,28,31,43,29,25,30,41)(21,37,39,42,38,34,44,40);(1,2,3,4)(5,8,11,12)(6,7,10,9)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,34)(22,39)(23,38)(24,40)(26,41)(27,28)(32,43)(33,44)(35,42)(36,37);(5,9,11,7)(6,12,10,8)(16,29)(21,38)(25,28)(30,31)(34,37)(39,44)(40,42)(41,43);(5,10,11,6)(7,8,9,12)(13,17,19,18)(14,22,23,33)(15,26,27,32)(16,30,29,31)(20,35,36,24)(21,39,38,44)(25,41,28,43)(34,42,37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477797);