Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ 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:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,43,29,32,30,35)(27,36,42,33,38,49,39,45)(28,34,44,54,40,46,41,48)(37,50,53,47,51,56,52,55);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(26,32)(33,45)(34,46)(35,43)(36,49)(47,55)(48,54)(50,56);(1,2)(3,8)(4,7)(5,6)(25,27,29,38)(26,33,32,45)(28,37,40,51)(30,42,31,39)(34,47,46,55)(35,49,43,36)(41,53,44,52)(48,56,54,50);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,34)(27,37)(29,40)(30,41)(31,44)(32,46)(33,47)(35,48)(36,50)(38,51)(39,52)(42,53)(43,54)(45,55)(49,56);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,32)(28,40)(33,45)(35,43)(36,49)(37,51)(41,44)(52,53)Polycyclic presentation:
2^9|a^2=fgh,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7198421);