Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,21,26)(19,28,29,23)(20,32,34,25)(22,24,36,27)(30,38,33,41)(31,37,43,39)(35,42,46,40)(44,48,45,47);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,23,26,28)(20,33,34,30)(22,31,36,43)(24,39,27,37)(25,41,32,38)(35,45,46,44)(40,47,42,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,24)(19,29)(22,36)(23,37)(25,40)(26,27)(28,39)(30,33)(32,42)(35,46)(38,47)(41,48);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,38)(24,40)(26,32)(27,42)(28,41)(29,33)(31,44)(36,46)(37,47)(39,48)(43,45);(9,11,12,15)(10,14,13,16)(19,29)(20,34)(23,28)(25,32)(31,43)(35,46)(37,39)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6561108);