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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,18)(13,21)(14,17)(16,22)(19,24)(20,23)(25,26,31,35)(27,32,36,29)(28,39,42,33)(30,34,46,50)(37,43,48,40)(38,47,45,44)(41,52,55,49)(51,56,54,53);(2,6)(3,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(27,36)(29,32)(37,48)(38,45)(40,43)(44,47)(51,54)(53,56);(1,3)(2,4)(5,7)(6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,27)(26,32)(28,37)(29,35)(30,45)(31,36)(33,40)(34,44)(38,46)(39,43)(41,54)(42,48)(47,50)(49,56)(51,55)(52,53);(1,4)(2,3)(5,8)(6,7)(25,28,31,42)(26,33,35,39)(27,37,36,48)(29,43,32,40)(30,41,46,55)(34,49,50,52)(38,51,45,54)(44,56,47,53);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,29,31,32)(26,27,35,36)(28,40,42,43)(30,44,46,47)(33,48,39,37)(34,38,50,45)(41,53,55,56)(49,54,52,51);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30)(26,34)(27,38)(28,41)(29,44)(31,46)(32,47)(33,49)(35,50)(36,45)(37,51)(39,52)(40,53)(42,55)(43,56)(48,54)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833980);