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 | #45101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1893 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)(15,23,29,21)(16,30,34,40)(17,36,28,38)(18,26,20,24)(22,33,41,35)(27,39,43,44)(31,37,32,42);(1,2)(3,4)(6,10)(11,12)(13,15,19,29)(14,21,25,23)(16,31,34,32)(17,18,28,20)(22,39,41,44)(24,38,26,36)(27,35,43,33)(30,42,40,37);(5,7,9,8)(6,11,10,12)(13,16,18,33,19,34,20,35)(14,22,24,40,25,41,26,30)(15,27,17,31,29,43,28,32)(21,37,36,39,23,42,38,44);(2,4)(13,17,20,15,19,28,18,29)(14,23,26,36,25,21,24,38)(16,32,35,43,34,31,33,27)(22,37,30,44,41,42,40,39);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,28,29,17)(16,33,34,35)(21,38,23,36)(22,40,41,30)(27,32,43,31)(37,44,42,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=g,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,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471800);