Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | |
1442 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,30)(29,43)(31,41)(32,42)(34,40)(35,48)(36,45)(37,46)(38,47)(39,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,41)(20,34,30,29)(21,36,38,35)(23,39,37,32)(25,43,33,40)(26,45,47,48)(28,44,46,42);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,32)(22,30)(23,35)(24,40)(26,44)(27,33)(28,45)(31,34)(36,37)(38,39)(41,43)(42,47)(46,48);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,37,29,22,38,23,34)(18,26,46,43,27,47,28,40)(19,30,39,35,31,20,32,36)(24,25,44,48,41,33,42,45);(2,6)(7,8)(20,30)(21,38)(25,33)(26,47)(29,34)(35,36)(40,43)(45,48)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517379);