Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 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)(3,7)(4,8)(5,6)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,40)(38,49)(39,51)(41,46)(42,50)(43,45)(44,48)(47,54)(52,56)(53,55);(2,6)(3,4)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(26,32)(27,38)(28,41)(35,36)(40,43)(42,44)(47,55)(51,56);(1,2)(3,8)(4,7)(5,6)(25,27,31,40,30,38,29,43)(26,33,36,37,32,45,35,49)(28,39,42,54,41,52,44,53)(34,47,48,51,46,55,50,56);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,34)(27,39)(29,44)(30,41)(31,42)(32,46)(33,47)(35,50)(36,48)(37,51)(38,52)(40,54)(43,53)(45,55)(49,56);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,29,30,31)(26,35,32,36)(27,40,38,43)(28,42,41,44)(33,37,45,49)(34,48,46,50)(39,53,52,54)(47,56,55,51)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7179170);