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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,32)(27,37,38,51)(28,34,41,47)(29,36,31,35)(33,44,46,40)(39,53,42,56)(43,52,45,50)(48,54,49,55);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,38)(28,41)(35,36)(40,44)(43,45)(48,49)(53,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,31,40,30,38,29,44)(26,33,36,37,32,46,35,51)(28,42,43,54,41,39,45,55)(34,49,50,56,47,48,52,53);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,39)(29,45)(30,41)(31,43)(32,47)(33,48)(35,52)(36,50)(37,53)(38,42)(40,55)(44,54)(46,49)(51,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,32,36)(27,40,38,44)(28,43,41,45)(33,37,46,51)(34,50,47,52)(39,54,42,55)(48,56,49,53)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6958769);