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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,5)(6,8)(10,14)(11,17)(12,20)(22,24)(25,26,31,36)(27,32,37,47)(28,33,42,39)(29,34,43,38)(30,45,46,35)(40,54,51,48)(41,53,55,49)(44,52,56,50);(2,5)(6,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(27,37)(29,43)(32,47)(34,38)(40,51)(44,56)(48,54)(50,52);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27)(26,32)(28,34)(29,39)(30,40)(31,37)(33,43)(35,48)(36,47)(38,42)(41,50)(44,53)(45,54)(46,51)(49,56)(52,55);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,33)(27,38)(29,32)(30,41)(31,42)(34,37)(35,49)(36,39)(40,52)(43,47)(44,48)(45,53)(46,55)(50,51)(54,56);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,34)(27,39)(28,32)(30,44)(31,43)(33,37)(35,50)(36,38)(40,53)(41,48)(42,47)(45,52)(46,56)(49,51)(54,55);(25,30,31,46)(26,35,36,45)(27,40,37,51)(28,41,42,55)(29,44,43,56)(32,48,47,54)(33,49,39,53)(34,50,38,52)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022719);