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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,12,19,15)(13,21,22,16)(18,23,24,20)(25,26,31,32)(27,33,38,46)(28,39,42,29)(30,44,45,34)(35,36,40,47)(37,51,52,48)(41,43,55,53)(49,56,54,50);(2,5)(6,7)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(26,32)(29,39)(33,46)(34,44)(36,47)(43,53)(48,51)(50,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,27)(26,33)(28,40)(29,36)(30,37)(31,38)(32,46)(34,48)(35,42)(39,47)(41,54)(43,50)(44,51)(45,52)(49,55)(53,56);(1,3)(2,6)(4,8)(5,7)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,28)(26,29)(27,35)(30,41)(31,42)(32,39)(33,47)(34,43)(36,46)(37,49)(38,40)(44,53)(45,55)(48,56)(50,51)(52,54);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29)(26,28)(27,36)(30,43)(31,39)(32,42)(33,40)(34,41)(35,46)(37,50)(38,47)(44,55)(45,53)(48,54)(49,51)(52,56);(25,30,31,45)(26,34,32,44)(27,37,38,52)(28,41,42,55)(29,43,39,53)(33,48,46,51)(35,49,40,54)(36,50,47,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^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, 8022600);