Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| 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 | #28349 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2032 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)(23,33)(24,38)(25,30)(26,31)(27,32)(28,35)(29,41)(34,42)(36,44)(37,43)(39,46)(40,45)(47,50)(48,52)(49,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,35,27,37,26,36)(22,28,31,33,32,44,30,43)(24,34,40,49,39,48,41,50)(29,42,46,47,45,52,38,51);(7,13)(8,15)(10,16)(11,18)(21,24,26,41,27,39,25,40)(22,29,30,38,32,45,31,46)(23,34,36,50,37,48,35,49)(28,42,43,51,44,52,33,47);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(24,39)(29,45)(34,48)(38,46)(40,41)(42,52)(47,51)(49,50);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,40,39,41)(28,43,44,33)(29,38,45,46)(34,49,48,50)(42,51,52,47)Polycyclic presentation:
2^9|a^2=h,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,e^b=eh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434159);