Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,26,30)(23,33,35,28)(24,29,38,46)(25,31,40,42)(27,32,37,44)(34,48,50,41)(36,45,51,43)(39,47,49,52);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,35)(22,28,30,33)(24,36,38,51)(25,27,40,37)(29,43,46,45)(31,44,42,32)(34,49,50,39)(41,52,48,47);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,37,49,26,38,27,39)(22,29,44,52,30,46,32,47)(23,34,40,51,35,50,25,36)(28,41,31,45,33,48,42,43);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(22,30)(24,38)(28,33)(31,42)(32,44)(34,50)(36,51)(39,49);(7,17)(11,14)(15,16)(19,20)(21,25,37,23,26,40,27,35)(22,31,44,33,30,42,32,28)(24,36,49,34,38,51,39,50)(29,45,52,48,46,43,47,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722712);