Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,15,12)(11,18,20,16)(13,21,14,17)(19,23,22,24)(25,26)(27,31)(28,33)(29,32)(30,35)(34,41)(36,44)(37,43)(38,45)(39,48)(40,46)(42,50)(47,52)(49,55)(51,54)(53,56);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,23)(21,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,45)(33,43)(34,42)(35,44)(39,49)(40,51)(41,50)(46,54)(47,53)(48,55)(52,56);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(26,32)(27,36)(28,39)(29,41)(31,42)(33,47)(34,35)(43,54)(44,45)(46,48)(51,56)(53,55);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28,30,39)(26,33,35,48)(27,37,36,49)(29,40,41,52)(31,43,44,55)(32,46,34,47)(38,51,50,56)(42,53,45,54);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(27,36)(28,39)(31,44)(33,48)(38,50)(40,52)(42,45)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fhi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6901020);