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 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,22,35)(20,30,32,37)(21,26,34,42)(25,36,40,28)(29,38,33,41)(31,44,46,47)(39,45,48,43);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,33,34,29)(25,37,40,30)(26,41,42,38)(31,45,46,43)(39,44,48,47);(3,7)(6,8)(11,16)(13,15)(17,20)(18,25)(19,28)(21,31)(22,36)(23,32)(24,37)(26,39)(27,40)(29,43)(30,35)(33,45)(34,46)(38,47)(41,44)(42,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,34)(18,26,27,42)(19,29,22,33)(20,31,32,46)(24,38,35,41)(25,39,40,48)(28,43,36,45)(30,44,37,47);(2,5)(6,8)(20,32)(21,34)(25,40)(26,42)(28,36)(29,33)(30,37)(38,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,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, 6837535);