Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,32)(23,33,35,44)(24,38,41,45)(25,30,26,31)(28,37,43,36)(29,39,46,40)(34,42,48,52)(47,49,51,50);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,36,27,35,26,37)(22,28,31,33,32,43,30,44)(24,34,39,49,41,48,40,50)(29,42,45,47,46,52,38,51);(7,13)(8,15)(10,16)(11,18)(21,24,26,40,27,41,25,39)(22,29,30,38,32,46,31,45)(23,34,37,50,35,48,36,49)(28,42,44,51,43,52,33,47);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(23,35)(28,43)(33,44)(34,48)(36,37)(42,52)(47,51)(49,50);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,44,43,33)(29,38,46,45)(34,49,48,50)(42,51,52,47)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434144);