Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #25614 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 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)(7,11)(10,12)(13,14)(15,21)(16,28)(17,23)(18,33)(19,24)(20,26)(22,31)(25,39)(27,36)(29,35)(30,41)(32,37)(34,40)(38,43)(42,44);(6,9)(7,11)(13,15)(14,20)(16,29)(17,26)(18,32)(19,27)(21,33)(22,39)(23,36)(24,37)(25,30)(28,38)(31,42)(34,43)(35,41)(40,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,25)(17,32)(18,26)(19,27)(20,33)(22,38)(23,37)(24,36)(28,39)(29,30)(31,43)(34,42)(35,41)(40,44);(6,9)(10,12)(13,16,19,31)(14,22,24,28)(15,25,27,43)(17,30,18,34)(20,35,37,44)(21,38,36,39)(23,40,33,41)(26,42,32,29);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,33)(15,26,27,32)(16,30,31,34)(20,36,37,21)(22,40,28,41)(25,42,43,29)(35,39,44,38)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055606);