Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1355 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,32)(29,43)(30,33)(31,39)(34,45)(36,53)(37,49)(38,54)(40,47)(41,51)(42,52)(44,50)(46,56)(48,55);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,39,30,38,45,55)(26,31,34,35,33,48,43,54)(28,40,42,37,41,36,44,56)(32,49,50,47,51,46,52,53);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,30,41)(26,32,33,51)(27,36,38,40)(29,42,45,44)(31,46,48,49)(34,50,43,52)(35,53,54,47)(37,39,56,55);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,32)(27,37)(29,44)(30,41)(31,47)(33,51)(34,52)(35,49)(36,55)(38,56)(39,40)(42,45)(43,50)(46,54)(48,53);(10,16)(11,18)(15,24)(17,19)(26,33)(27,38)(28,41)(34,43)(39,55)(42,44)(46,49)(47,53)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=h,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7018863);