Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1416 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,34)(22,25)(23,27)(28,38)(29,37)(30,36)(31,39)(32,42)(33,40)(35,41)(43,48)(44,47)(45,46);(1,3)(2,6)(4,7)(5,8)(17,19,22,31)(18,24,25,39)(20,29,33,44)(21,30,23,28)(26,37,40,47)(27,38,34,36)(32,43,35,45)(41,46,42,48);(9,11,12,16)(10,13,14,15)(18,25)(19,28)(21,23)(24,36)(26,40)(29,43)(30,31)(32,35)(37,46)(38,39)(44,45)(47,48);(2,7)(3,8)(10,14)(13,15)(17,20)(18,26)(19,29)(21,35)(22,33)(23,32)(24,37)(25,40)(27,42)(28,43)(30,45)(31,44)(34,41)(36,46)(38,48)(39,47);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,25,34)(19,30,31,28)(20,32,33,35)(24,38,39,36)(26,41,40,42)(29,43,44,45)(37,46,47,48)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7415859);