Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,26)(19,29,31,39,32,42,37,24)(20,33,36,40,34,25,35,41)(30,44,46,38,45,43,48,47);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,29)(27,42)(28,39)(33,47)(34,45)(35,46)(36,48)(40,43)(41,44);(2,6)(5,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,34)(24,38)(26,40)(27,41)(28,33)(29,43)(31,46)(32,45)(37,48)(39,47)(42,44);(9,11)(10,13)(12,15)(14,16)(20,34)(25,33)(30,45)(35,36)(38,47)(40,41)(43,44)(46,48);(11,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,34,36)(24,29,39,42)(25,40,33,41)(30,46,45,48)(38,43,47,44)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6512881);