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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(6,8)(9,10,14,17)(11,12,19,15)(13,21,22,16)(18,23,24,20)(25,26,31,35)(27,32,36,29)(28,39,40,33)(30,34,46,50)(37,43,48,41)(38,47,45,44)(42,52,53,49)(51,56,55,54);(3,7)(6,8)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(27,36)(28,40)(29,32)(33,39)(38,45)(42,53)(44,47)(49,52);(1,2)(3,6)(4,5)(7,8)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,27)(26,32)(28,37)(29,35)(30,45)(31,36)(33,41)(34,44)(38,46)(39,43)(40,48)(42,55)(47,50)(49,56)(51,53)(52,54);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,28,31,40)(26,33,35,39)(27,37,36,48)(29,43,32,41)(30,42,46,53)(34,49,50,52)(38,51,45,55)(44,56,47,54);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29,31,32)(26,27,35,36)(28,41,40,43)(30,44,46,47)(33,48,39,37)(34,38,50,45)(42,54,53,56)(49,55,52,51);(25,30)(26,34)(27,38)(28,42)(29,44)(31,46)(32,47)(33,49)(35,50)(36,45)(37,51)(39,52)(40,53)(41,54)(43,56)(48,55)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324083);