Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1296 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)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,35)(28,33)(29,31)(30,43)(32,39)(34,36)(37,50)(38,49)(40,45)(41,54)(42,56)(44,51)(46,53)(47,55)(48,52);(1,2,4,5)(3,7,8,6)(11,17)(12,20)(14,21)(15,19)(26,31)(28,40)(32,44)(34,43)(35,49)(37,41)(42,52)(53,55);(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,36,51,29,38,30,39)(26,32,43,49,31,44,34,35)(28,41,52,53,40,37,42,55)(33,47,56,50,45,46,48,54);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,37)(29,40)(30,42)(31,45)(32,46)(34,48)(35,50)(36,52)(38,41)(39,53)(43,56)(44,47)(49,54)(51,55);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,40)(33,45)(37,41)(42,52)(46,47)(48,56)(50,54)(53,55)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=ghi,d^2=fi,e^2=h,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104195);