Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1278 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,33)(27,35,38,51)(28,32,41,49)(29,43,36,34)(31,44,46,39)(37,50,40,54)(42,56,52,48)(45,55,47,53);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,36,44,30,38,29,39)(26,31,43,51,33,46,34,35)(28,40,52,53,41,37,42,55)(32,47,56,50,49,45,48,54);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,30,41)(26,32,33,49)(27,37,38,40)(29,42,36,52)(31,45,46,47)(34,48,43,56)(35,50,51,54)(39,53,44,55);(11,18)(14,22)(17,19)(21,23)(25,28)(26,32)(27,37)(29,42)(30,41)(31,45)(33,49)(34,48)(35,50)(36,52)(38,40)(39,53)(43,56)(44,55)(46,47)(51,54);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(26,33)(27,38)(28,41)(34,43)(39,44)(42,52)(45,47)(50,54)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di,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, 6388485);