Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 52 points:
(2,4)(6,10)(8,16)(11,13)(14,20)(21,22,27,32)(23,24,29,28)(25,37,40,30)(26,41,42,31)(33,43,38,35)(34,44,45,36)(39,46,51,49)(47,48,50,52);(1,2,3,4)(23,29)(24,28)(33,38)(34,45)(35,43)(36,44)(47,50)(48,52);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23)(22,28)(24,32)(25,38)(26,34)(27,29)(30,35)(31,44)(33,40)(36,41)(37,43)(39,50)(42,45)(46,48)(47,51)(49,52);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,27,28)(22,29,32,23)(25,35,40,43)(26,36,42,44)(30,38,37,33)(31,45,41,34)(39,48,51,52)(46,50,49,47);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,27,40)(22,30,32,37)(23,33,29,38)(24,35,28,43)(26,39,42,51)(31,46,41,49)(34,47,45,50)(36,48,44,52);(21,26,27,42)(22,31,32,41)(23,34,29,45)(24,36,28,44)(25,39,40,51)(30,46,37,49)(33,47,38,50)(35,48,43,52)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622746);