Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,23)(16,30)(17,36)(18,24)(19,26)(20,25)(21,28)(22,34)(27,43)(29,38)(31,40)(32,37)(33,42)(35,41)(39,44);(1,2)(3,4)(6,10)(11,12)(13,15,18,28)(14,21,24,23)(16,31,33,32)(17,20,29,19)(22,39,41,43)(25,36,26,38)(27,35,44,34)(30,37,42,40);(5,7,9,8)(6,11,10,12)(13,16,19,34,18,33,20,35)(14,22,25,30,24,41,26,42)(15,27,29,31,28,44,17,32)(21,37,38,39,23,40,36,43);(2,4)(5,7,9,8)(6,11,10,12)(13,17,20,28,18,29,19,15)(14,23,26,38,24,21,25,36)(16,32,35,44,33,31,34,27)(22,40,42,39,41,37,30,43);(5,8,9,7)(6,11,10,12)(14,24)(16,33)(21,23)(25,26)(27,44)(31,32)(34,35)(36,38)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472651);