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 | |
6258 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)(5,6)(7,15)(8,14)(9,12)(10,16)(11,13)(17,18)(19,20)(23,31)(28,32)(33,43)(34,44)(40,45)(41,46)(47,51)(50,52);(1,2)(3,4)(5,7,11,9)(6,12,13,15)(8,17,16,19)(10,18,14,20)(22,27)(25,38)(28,32)(29,35)(34,44)(36,49)(40,45)(47,51);(6,13)(8,16)(12,15)(17,19)(21,22,26,27)(23,32,31,28)(24,35,37,29)(25,30,38,39)(33,40,43,45)(34,46,44,41)(36,48,49,42)(47,50,51,52);(5,8,11,16)(6,14,13,10)(7,17,9,19)(12,20,15,18)(21,23,26,31)(22,28,27,32)(24,33,37,43)(25,34,38,44)(29,40,35,45)(30,41,39,46)(36,47,49,51)(42,50,48,52);(5,9,11,7)(6,15,13,12)(8,19,16,17)(10,20,14,18)(21,24,26,37)(22,29,27,35)(23,33,31,43)(25,36,38,49)(28,40,32,45)(30,42,39,48)(34,47,44,51)(41,50,46,52);(5,10,11,14)(6,16,13,8)(7,18,9,20)(12,19,15,17)(21,25)(22,30)(23,34)(24,36)(26,38)(27,39)(28,41)(29,42)(31,44)(32,46)(33,47)(35,48)(37,49)(40,50)(43,51)(45,52)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650661);