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 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,31,35)(27,32,36,29)(28,39,43,33)(30,34,47,50)(37,44,40,41)(38,48,46,45)(42,52,55,49)(51,56,53,54);(1,3,5,8)(2,6,7,4)(10,15)(11,18)(17,20)(19,24)(27,36)(29,32)(37,40)(38,46)(41,44)(45,48)(51,53)(54,56);(1,3)(2,4)(5,8)(6,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,27)(26,32)(28,40)(29,35)(30,46)(31,36)(33,44)(34,45)(37,43)(38,47)(39,41)(42,51)(48,50)(49,54)(52,56)(53,55);(1,4)(2,3)(5,6)(7,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28,31,43)(26,33,35,39)(27,37,36,40)(29,44,32,41)(30,42,47,55)(34,49,50,52)(38,51,46,53)(45,56,48,54);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,29,31,32)(26,27,35,36)(28,41,43,44)(30,45,47,48)(33,40,39,37)(34,38,50,46)(42,54,55,56)(49,53,52,51);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,30)(26,34)(27,38)(28,42)(29,45)(31,47)(32,48)(33,49)(35,50)(36,46)(37,51)(39,52)(40,53)(41,54)(43,55)(44,56)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839718);