Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2828 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,26,30,25,29,35,42)(23,27,33,40,32,39,45,50)(24,34,37,48,36,28,41,43)(31,44,47,52,46,38,49,51);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23)(22,27)(24,31)(25,32)(26,33)(28,38)(29,39)(30,40)(34,44)(35,45)(36,46)(37,47)(41,49)(42,50)(43,51)(48,52);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,35,41,25,36,26,37)(22,28,42,48,29,34,30,43)(23,31,45,49,32,46,33,47)(27,38,50,52,39,44,40,51);(7,16)(11,14)(15,17)(19,20)(22,29)(23,32)(28,34)(30,42)(31,46)(33,45)(43,48)(47,49);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(23,32)(27,39)(31,46)(33,45)(38,44)(40,50)(47,49)(51,52)Polycyclic presentation:
2^9|a^2=h,c^2=fgh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301106);