Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(23,32)(24,29)(25,30)(26,41)(27,31)(28,35)(33,47)(34,43)(36,48)(37,51)(38,45)(39,46)(40,52)(42,50)(44,49);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,34,27,36,26,35)(22,28,41,48,31,43,30,32)(24,37,39,50,38,33,40,49)(29,44,52,47,45,42,46,51);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,40)(27,38)(28,42)(30,46)(31,45)(32,47)(34,49)(35,50)(36,37)(41,52)(43,44)(48,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,38)(29,45)(33,37)(39,40)(42,44)(46,52)(47,51)(49,50);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,39,38,40)(28,32,43,48)(29,46,45,52)(33,49,37,50)(42,47,44,51)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6471792);