Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 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)(5,6,8,9)(7,11,12,10)(13,14)(15,23)(16,27)(17,21)(18,22)(19,26)(20,29)(24,34)(25,39)(28,36)(30,40)(31,38)(32,35)(33,37)(41,42)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,28)(17,26)(18,32)(20,36)(21,23)(22,39)(24,30)(25,31)(27,44)(29,43)(33,42)(34,37)(35,38)(40,41);(1,2,3,4)(6,9)(10,11)(13,16)(14,20)(15,24)(17,29)(18,33)(19,34)(21,27)(22,40)(23,41)(25,43)(26,42)(28,32)(30,31)(35,44)(36,39)(37,38);(5,7,8,12)(6,10,9,11)(14,21)(16,29)(19,23)(22,38)(24,42)(28,43)(30,33)(35,39);(15,25)(18,31)(19,35)(22,38)(23,39)(24,43)(26,32)(28,42)(30,33)(34,44)(36,41)(37,40)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591924);