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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1868 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)(7,9)(8,10)(11,12)(14,20)(15,25)(17,18)(21,36)(22,29)(23,24)(26,35)(27,30)(31,32)(37,41)(38,44)(39,40);(5,6)(7,11)(8,10)(9,12)(13,14,17,23,19,24,18,20)(15,26,34,35,25,21,28,36)(16,29,31,40,33,39,32,22)(27,41,43,37,30,44,42,38);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,34)(18,28)(19,25)(20,35)(22,38)(23,36)(24,26)(27,33)(29,41)(31,42)(32,43)(37,40)(39,44);(7,12)(9,11)(13,16,19,33)(14,22,24,40)(15,27,25,30)(17,32,18,31)(20,29,23,39)(21,37,26,38)(28,43,34,42)(35,44,36,41);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,28,25,34)(16,31,33,32)(21,35,26,36)(22,29,40,39)(27,42,30,43)(37,41,38,44)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910798);