Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(17,18,23,37,22,28,21,27)(19,24,31,40,33,42,32,41)(20,26,34,25,36,44,35,45)(29,46,48,38,30,39,47,43);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,42)(20,30,36,29)(21,32,23,31)(25,43,45,38)(26,39,44,46)(27,41,37,40)(34,47,35,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,45)(19,29,33,30)(21,34,23,35)(24,38,42,43)(26,27,44,37)(31,47,32,48)(39,40,46,41);(11,16)(13,15)(17,20)(18,26)(19,30)(21,35)(22,36)(23,34)(24,39)(25,37)(27,45)(28,44)(29,33)(31,47)(32,48)(38,41)(40,43)(42,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,31,33,32)(20,34,36,35)(24,40,42,41)(25,44,45,26)(29,47,30,48)(38,46,43,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413164);