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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1299 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)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,34)(29,35)(30,32)(31,45)(33,38)(37,53)(39,49)(40,47)(41,56)(42,51)(43,46)(44,52)(48,55)(50,54);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(28,40)(33,46)(35,45)(36,49)(37,41)(42,44)(54,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,43,30,39,31,38)(26,33,45,49,32,46,35,36)(28,41,44,55,40,37,42,54)(34,50,51,53,47,48,52,56);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,40)(26,34,32,47)(27,37,39,41)(29,44,31,42)(33,48,46,50)(35,52,45,51)(36,53,49,56)(38,55,43,54);(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,32,45)(27,38,39,43)(28,42,40,44)(33,49,46,36)(34,51,47,52)(37,54,41,55)(48,53,50,56)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6950180);