Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1379 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)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,30,32)(27,37,40,50)(28,41,42,34)(29,36,31,35)(33,45,46,39)(38,52,53,51)(43,48,44,49)(47,55,56,54);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(26,32)(28,42)(33,46)(35,36)(37,50)(38,53)(43,44)(54,55);(3,7)(5,8)(25,27,30,40)(26,33,32,46)(28,38,42,53)(29,39,31,45)(34,47,41,56)(35,37,36,50)(43,54,44,55)(48,52,49,51);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,41)(33,47)(35,49)(36,48)(37,51)(39,55)(40,53)(45,54)(46,56)(50,52);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,29,30,31)(26,35,32,36)(27,39,40,45)(28,43,42,44)(33,37,46,50)(34,48,41,49)(38,54,53,55)(47,52,56,51)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7113712);