Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | |
1517 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)(3,7)(4,6)(5,8)(17,18,22,26)(19,27,29,41)(20,24,31,38)(21,33,34,25)(23,35,37,30)(28,40,42,48)(32,45,44,39)(36,46,47,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(40,48)(43,46);(2,6)(3,8)(10,13)(14,16)(17,19,21,30,22,29,34,35)(18,23,25,27,26,37,33,41)(20,28,32,43,31,42,44,46)(24,36,39,40,38,47,45,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,39)(26,38)(27,40)(29,42)(30,43)(33,45)(34,44)(35,46)(37,47)(41,48);(10,13)(14,16)(19,29)(20,31)(23,37)(24,38)(27,41)(30,35)(32,44)(39,45)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693552);