Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,23,28)(19,24,33,41)(20,26,35,43)(21,27,22,37)(25,36,44,34)(29,45,30,39)(31,40,32,48)(38,47,42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,42)(26,39)(27,40)(28,41)(29,35)(34,47)(36,46)(37,48)(38,44)(43,45);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,43)(28,44)(30,33)(31,46)(32,47)(39,40)(41,42)(45,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,39)(25,37)(27,44)(28,43)(29,33)(31,46)(32,47)(38,40)(41,45)(42,48);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,34,35,36)(24,40,41,48)(25,43,44,26)(29,46,30,47)(38,39,42,45)Polycyclic presentation:
2^9|a^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6674355);