Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14,17,19)(15,23,26,36)(16,27,29,38)(18,31,30,22)(20,32,34,24)(21,33,35,28)(25,37,41,43)(39,40,44,42);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(15,24)(16,28)(18,30)(20,36)(21,38)(23,34)(26,32)(27,35)(29,33)(37,43)(40,42);(13,15,17,26)(14,20,19,34)(16,25,29,41)(18,32,30,24)(21,37,35,43)(22,36,31,23)(27,39,38,44)(28,40,33,42);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,29)(14,21,19,35)(15,25,26,41)(18,33,30,28)(20,37,34,43)(22,38,31,27)(23,39,36,44)(24,40,32,42);(5,8,9,7)(6,11,12,10)(14,19)(15,26)(21,35)(22,31)(24,32)(25,41)(27,38)(40,42)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=f,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=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430400);