Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,32,30,36,29,35)(27,37,40,48,38,33,41,46)(28,34,44,47,45,52,42,51)(39,50,53,56,43,49,54,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,38)(28,42)(29,31)(33,48)(35,36)(37,46)(39,53)(43,54)(44,45)(47,51)(49,50);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,33)(28,43)(29,41)(30,38)(31,40)(32,46)(34,50)(35,48)(36,37)(39,45)(42,53)(44,54)(47,56)(49,52)(51,55);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,42,30,45,31,44)(26,34,35,51,36,52,32,47)(27,39,41,54,38,43,40,53)(33,49,48,56,37,50,46,55);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,36,32)(27,40,38,41)(28,44,45,42)(33,46,37,48)(34,47,52,51)(39,54,43,53)(49,56,50,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=gh,d^2=fgi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031033);