Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 |
| prime radical | #28452 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2093 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)(5,6,11,10)(7,8,9,12)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,26)(25,38)(27,34)(28,36)(29,37)(30,42)(31,35)(33,40)(39,44)(41,43);(5,7,11,9)(6,12,10,8)(13,15,18,26,19,27,17,31)(14,20,22,24,23,35,32,34)(16,25,30,40,28,39,29,41)(21,33,37,38,36,43,42,44);(1,2,3,4)(5,8,11,12)(6,7,10,9)(13,16,19,28)(14,21,23,36)(15,25,27,39)(17,29,18,30)(20,33,35,43)(22,37,32,42)(24,38,34,44)(26,40,31,41);(5,9,11,7)(6,12,10,8)(16,28)(21,36)(25,39)(29,30)(33,43)(37,42)(38,44)(40,41);(5,10,11,6)(7,8,9,12)(13,17,19,18)(14,22,23,32)(15,26,27,31)(16,29,28,30)(20,34,35,24)(21,37,36,42)(25,40,39,41)(33,44,43,38)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477806);