Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 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)(8,16)(11,17)(15,20)(18,19)(21,22)(23,31)(24,35)(25,39)(26,29)(27,33)(28,37)(30,40)(32,47)(34,46)(36,45)(38,44)(41,49)(42,51)(43,50)(48,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,25,33,26,34,40,51)(22,27,30,31,29,42,39,46)(24,36,38,49,37,32,48,50)(28,43,44,47,35,41,52,45);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,28)(23,32)(25,38)(26,37)(27,41)(29,35)(30,44)(31,45)(33,50)(34,36)(39,52)(40,48)(42,43)(46,47)(49,51);(7,14)(10,13)(15,20)(18,19)(23,33)(24,37)(25,40)(27,31)(28,35)(30,39)(32,49)(34,51)(36,50)(41,47)(42,46)(43,45);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,29)(23,34)(24,37)(30,39)(33,51)(38,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089934);