Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207830 |
| 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 | #53073 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1283 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,35)(28,33)(29,43)(30,31)(32,37)(34,44)(36,52)(38,51)(39,46)(40,56)(41,50)(42,49)(45,55)(47,54)(48,53);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(26,31)(28,39)(32,45)(34,43)(35,51)(36,40)(41,42)(53,54);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,37,30,38,44,55)(26,32,34,35,31,45,43,51)(28,40,42,53,39,36,41,54)(33,48,50,56,46,47,49,52);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28,30,39)(26,33,31,46)(27,36,38,40)(29,42,44,41)(32,47,45,48)(34,50,43,49)(35,52,51,56)(37,54,55,53);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,44)(26,34,31,43)(27,37,38,55)(28,41,39,42)(32,35,45,51)(33,49,46,50)(36,53,40,54)(47,56,48,52)Polycyclic presentation:
2^9|b^2=fi,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118794);