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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 52 points:
(1,2,3,4)(6,10)(7,13)(12,17)(14,19)(21,22,26,30)(23,31,32,27)(24,28,37,35)(25,38,39,29)(33,43,45,40)(34,41,46,44)(36,48,49,42)(47,50,52,51);(1,2)(3,4)(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(23,32)(27,31)(33,45)(34,46)(40,43)(41,44)(47,52)(50,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,35)(25,29)(26,30)(28,37)(31,32)(33,43)(34,41)(36,48)(38,39)(40,45)(42,49)(44,46)(47,51)(50,52);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,27)(24,33)(25,34)(26,32)(28,40)(29,41)(30,31)(35,43)(36,47)(37,45)(38,44)(39,46)(42,50)(48,51)(49,52);(21,24)(22,28)(23,33)(25,36)(26,37)(27,40)(29,42)(30,35)(31,43)(32,45)(34,47)(38,48)(39,49)(41,50)(44,51)(46,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,39)(22,29,30,38)(23,34,32,46)(24,36,37,49)(27,41,31,44)(28,42,35,48)(33,47,45,52)(40,50,43,51)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551252);