Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632074 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,14,18)(11,19,21,15)(12,17,13,16)(20,23,22,24)(25,26,31,37)(27,38,41,33)(28,35,29,44)(30,36,32,47)(34,46,51,43)(39,49,45,52)(40,54,42,50)(48,55,53,56);(1,2,4,5)(3,6,7,8)(9,11,14,21)(10,15,18,19)(12,22,13,20)(16,24,17,23)(25,27)(26,33)(28,39)(29,45)(30,40)(31,41)(32,42)(34,48)(35,52)(36,50)(37,38)(43,55)(44,49)(46,56)(47,54)(51,53);(2,5)(6,8)(25,28,31,29)(26,34,37,51)(27,39,41,45)(30,43,32,46)(33,48,38,53)(35,36,44,47)(40,55,42,56)(49,54,52,50);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,29,31,28)(26,35,37,44)(27,39,41,45)(30,43,32,46)(33,49,38,52)(34,36,51,47)(40,56,42,55)(48,54,53,50);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,31,32)(26,36,37,47)(27,40,41,42)(28,43,29,46)(33,50,38,54)(34,44,51,35)(39,55,45,56)(48,49,53,52)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6565756);