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 | |
1340 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)(11,17,18,14)(12,20,22,15)(19,23,21,24)(25,26)(27,31)(28,39)(29,33)(30,35)(32,36)(34,38)(37,45)(40,55)(41,44)(42,56)(43,49)(46,51)(47,52)(48,53)(50,54);(1,3)(2,6)(4,7)(5,8)(9,11,13,18)(10,14,16,17)(12,21,22,19)(15,24,20,23)(25,27,30,38)(26,31,35,34)(28,36,41,52)(29,37,43,54)(32,44,47,39)(33,45,49,50)(40,51,42,53)(46,56,48,55);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,24)(21,22)(25,28,38,52,30,41,27,36)(26,32,31,44,35,47,34,39)(29,42,54,51,43,40,37,53)(33,48,45,55,49,46,50,56);(3,8)(6,7)(11,18)(14,17)(19,21)(23,24)(25,29,30,43)(26,33,35,49)(27,37,38,54)(28,40,41,42)(31,45,34,50)(32,46,47,48)(36,51,52,53)(39,55,44,56);(9,12,13,22)(10,15,16,20)(11,19,18,21)(14,23,17,24)(28,41)(29,43)(32,47)(33,49)(36,52)(37,54)(39,44)(45,50)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fgi,d^2=i,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093539);