Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ 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:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,26)(16,22)(17,33)(18,24)(19,25)(20,23)(21,30)(27,44)(28,38)(29,42)(31,39)(32,43)(34,37)(35,41)(36,40);(5,7,10,6)(8,12,9,11)(13,15)(14,21)(16,18)(17,28)(19,30)(20,36)(22,41)(23,38)(24,43)(25,26)(27,29)(31,34)(32,35)(33,40)(37,42)(39,44);(2,4)(13,16,17,31,19,32,20,29)(14,22,33,39,25,43,23,42)(15,27,36,35,30,34,28,18)(21,37,38,24,26,44,40,41);(5,8,10,9)(6,11,7,12)(13,17,19,20)(14,23,25,33)(15,28,30,36)(16,31,32,29)(18,34,35,27)(21,38,26,40)(22,42,43,39)(24,44,41,37);(5,9,10,8)(6,11,7,12)(13,18,17,27,19,35,20,34)(14,24,33,44,25,41,23,37)(15,29,36,16,30,31,28,32)(21,39,38,43,26,42,40,22)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718742);