Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18,22,24)(19,28,29,45)(20,33,36,26)(21,27,23,38)(25,31,39,32)(30,46,34,44)(35,43,37,42)(40,47,41,48);(1,2,6,3)(4,7,5,8)(9,10,12,13)(11,14,15,16)(18,24)(19,29)(26,33)(27,38)(30,34)(31,32)(42,43)(47,48);(1,3,6,2)(4,7,5,8)(10,13)(14,16)(17,19,21,31,22,29,23,32)(18,25,38,45,24,39,27,28)(20,34,37,47,36,30,35,48)(26,41,42,44,33,40,43,46);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,37)(22,36)(23,35)(24,33)(25,40)(27,43)(28,44)(29,34)(31,48)(32,47)(38,42)(39,41)(45,46);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,27,24,38)(19,31,29,32)(20,35,36,37)(25,28,39,45)(26,42,33,43)(30,47,34,48)(40,46,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6878716);