Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | #45248 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14,19,25)(15,23,28,21)(16,30,33,41)(17,35,29,37)(18,26,20,24)(22,34,40,32)(27,38,42,44)(31,36,43,39);(1,2)(3,4)(6,10)(11,12)(13,15,19,28)(14,21,25,23)(16,31,33,43)(17,20,29,18)(22,38,40,44)(24,35,26,37)(27,32,42,34)(30,39,41,36);(5,7,9,8)(6,11,10,12)(13,16,20,34,19,33,18,32)(14,22,26,41,25,40,24,30)(15,27,17,31,28,42,29,43)(21,36,35,38,23,39,37,44);(2,4)(5,7,9,8)(6,11,10,12)(13,17,20,28,19,29,18,15)(14,23,26,37,25,21,24,35)(16,31,34,42,33,43,32,27)(22,39,41,44,40,36,30,38);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,17,28,29)(16,32,33,34)(21,35,23,37)(22,30,40,41)(27,31,42,43)(36,38,39,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fg,e^2=f,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7473000);