Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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×#3326 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1899 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)(7,11)(8,12)(9,10)(13,14,18,23)(15,25,28,20)(16,30,34,21)(17,35,32,22)(19,24,33,41)(26,37,44,43)(27,38,31,40)(29,42,36,39);(6,10)(11,12)(13,15)(14,20)(16,31)(17,26)(18,28)(19,36)(21,40)(22,37)(23,25)(24,42)(27,34)(29,33)(30,38)(32,44)(35,43)(39,41);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,26)(17,33)(18,34)(19,32)(20,37)(22,41)(23,30)(24,35)(25,43)(27,29)(28,44)(31,36)(38,39)(40,42);(5,8,9,7)(6,12,10,11)(16,32)(17,34)(19,33)(21,35)(22,30)(24,41)(26,27)(29,36)(31,44)(37,38)(39,42)(40,43);(13,17,18,32)(14,22,23,35)(15,27,28,31)(16,33,34,19)(20,38,25,40)(21,41,30,24)(26,29,44,36)(37,39,43,42)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491705);