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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1777 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)(15,21)(16,28)(17,33)(18,24)(19,36)(20,25)(22,31)(23,27)(26,42)(29,41)(30,40)(32,44)(34,37)(35,38)(39,43);(1,2)(3,4)(6,10)(11,12)(13,15,20,27)(14,21,25,23)(16,29,32,34)(17,19,35,18)(22,39,40,42)(24,33,36,38)(26,31,43,30)(28,41,44,37);(5,7,9,8)(6,11,10,12)(13,16,19,31,20,32,18,30)(14,22,24,28,25,40,36,44)(15,26,17,34,27,43,35,29)(21,37,38,42,23,41,33,39);(2,4)(13,17,19,27,20,35,18,15)(14,23,24,33,25,21,36,38)(16,29,31,26,32,34,30,43)(22,37,28,42,40,41,44,39);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,25,36)(15,17,27,35)(16,30,32,31)(21,33,23,38)(22,28,40,44)(26,34,43,29)(37,39,41,42)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=g,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=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, 7471192);