Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(10,14)(15,18)(16,20)(22,24)(25,26,32,35)(27,29,28,38)(30,33,44,49)(31,45,46,34)(36,41,39,47)(37,51,40,42)(43,54,55,48)(50,56,52,53);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,32,28)(26,29,35,38)(30,39,44,36)(31,37,46,40)(33,47,49,41)(34,42,45,51)(43,52,55,50)(48,56,54,53);(2,6)(3,7)(11,17)(15,18)(19,23)(22,24)(25,28,32,27)(26,29,35,38)(30,39,44,36)(31,40,46,37)(33,41,49,47)(34,42,45,51)(43,52,55,50)(48,53,54,56);(3,7)(4,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,29)(26,28)(27,35)(30,41)(31,42)(32,38)(33,39)(34,40)(36,49)(37,45)(43,53)(44,47)(46,51)(48,52)(50,54)(55,56);(1,3)(2,4)(5,7)(6,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,30,32,44)(26,33,35,49)(27,36,28,39)(29,41,38,47)(31,43,46,55)(34,48,45,54)(37,50,40,52)(42,53,51,56);(1,4)(2,3)(5,8)(6,7)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,31)(26,34)(27,37)(28,40)(29,42)(30,43)(32,46)(33,48)(35,45)(36,50)(38,51)(39,52)(41,53)(44,55)(47,56)(49,54)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532654);