Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(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)(29,39)(34,42)(36,44)(37,43)(40,45)(41,46)(47,48)(49,52)(50,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,27,37,25,35)(22,28,30,43,32,44,31,33)(24,34,39,48,40,49,41,50)(29,42,38,51,46,52,45,47);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,25,41,27,40,26,39)(22,29,31,45,32,46,30,38)(23,34,35,50,37,49,36,48)(28,42,33,47,44,52,43,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,40,41)(28,43,44,33)(29,45,46,38)(34,48,49,50)(42,47,52,51);(8,15)(11,18)(14,19)(17,20)(24,40)(29,46)(34,49)(38,45)(39,41)(42,52)(47,51)(48,50)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473376);