Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | #37379 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1533 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)(19,28)(20,32)(21,27)(22,24)(23,37)(25,30)(26,35)(29,46)(31,41)(33,45)(34,44)(36,43)(38,39)(40,47)(42,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(18,24)(25,39)(26,32)(27,37)(28,41)(40,42)(43,44)(45,46);(3,8)(5,7)(10,13)(11,15)(17,19)(18,25)(20,33)(21,30)(22,31)(23,38)(24,39)(26,42)(27,41)(28,37)(29,35)(32,40)(34,48)(36,47)(43,46)(44,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,29)(21,36)(22,35)(23,34)(24,32)(25,40)(27,44)(28,45)(30,48)(31,33)(37,43)(38,47)(39,42)(41,46);(10,13)(14,16)(17,21,22,23)(18,27,24,37)(19,30,31,38)(20,34,35,36)(25,41,39,28)(26,43,32,44)(29,47,33,48)(40,45,42,46)Polycyclic presentation:
2^9|a^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6837688);