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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(6,10)(8,11)(14,19)(15,23)(16,27)(17,31)(20,36)(21,28)(22,32)(24,34)(35,39)(37,41)(38,44)(42,43);(5,6,9,10)(7,8,12,11)(13,14)(15,24)(16,28)(17,32)(18,22)(19,31)(20,23)(21,30)(25,41)(26,34)(27,35)(29,44)(33,39)(36,40)(37,43)(38,42);(2,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,20)(16,29)(17,26)(18,23)(19,34)(21,38)(22,24)(25,27)(28,41)(30,43)(31,40)(32,36)(33,42)(35,44)(37,39);(6,10)(7,12)(13,16,18,27)(14,21,22,39)(15,25,23,29)(17,33,31,30)(19,35,32,28)(20,37,24,38)(26,43,40,42)(34,41,36,44);(5,8,9,11)(6,7,10,12)(14,22)(15,23)(19,32)(21,39)(25,29)(26,40)(28,35)(42,43)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 7089400);