Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1407 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,37,32,38,30,36)(27,33,41,49,42,50,46,52)(28,43,44,51,29,35,45,34)(39,53,55,56,40,48,54,47);(1,3,4,8)(2,5,6,7)(10,14)(15,20)(16,21)(22,24)(25,27)(26,33)(28,39)(29,40)(30,46)(31,41)(32,42)(34,47)(35,48)(36,52)(37,49)(38,50)(43,53)(44,55)(45,54)(51,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,32,29)(26,34,38,51)(27,39,42,40)(30,44,31,45)(33,47,50,56)(35,36,43,37)(41,54,46,55)(48,52,53,49);(2,6)(5,7)(11,17)(15,20)(18,23)(22,24)(25,29,32,28)(26,35,38,43)(27,40,42,39)(30,44,31,45)(33,48,50,53)(34,36,51,37)(41,54,46,55)(47,52,56,49);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,32,31)(26,36,38,37)(27,41,42,46)(28,44,29,45)(33,49,50,52)(34,43,51,35)(39,54,40,55)(47,48,56,53)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,e^b=ei,d^c=dfh,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, 6539083);