Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1992 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,40)(20,31,33,41)(21,26,22,36)(25,35,42,32)(28,43,46,48)(29,38,34,39)(37,44,47,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,39)(27,40)(31,43)(32,45)(33,46)(35,44)(36,38)(41,48)(42,47);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,21,35,23,33,22,32)(18,25,36,41,27,42,26,31)(19,28,34,44,30,46,29,45)(24,37,38,48,40,47,39,43);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,34)(20,32,33,35)(24,38,40,39)(25,41,42,31)(28,44,46,45)(37,43,47,48);(3,8)(5,7)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,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=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815112);