Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,5)(3,11,7,9)(4,13,6,10)(12,15,14,16)(20,28)(23,30)(27,32)(29,31)(33,34,40,45)(35,41,47,38)(36,44,39,42)(37,48,46,43);(1,3,8,7)(2,9,5,11)(4,12,6,14)(10,15,13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,35)(34,41)(36,37)(38,45)(39,46)(40,47)(42,43)(44,48);(1,4,8,6)(2,10,5,13)(3,12,7,14)(9,15,11,16)(17,19)(18,21)(20,27)(22,25)(23,29)(24,26)(28,32)(30,31)(33,36,40,39)(34,42,45,44)(35,46,47,37)(38,43,41,48);(1,5,8,2)(3,11,7,9)(4,10,6,13)(12,15,14,16)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,32)(26,30)(33,37,40,46)(34,43,45,48)(35,36,47,39)(38,44,41,42);(1,6,8,4)(2,10,5,13)(3,12,7,14)(9,16,11,15)(17,19)(18,24)(20,27)(21,26)(22,25)(23,31)(28,32)(29,30)(33,38)(34,35)(36,43)(37,44)(39,48)(40,41)(42,46)(45,47);(1,7,8,3)(2,11,5,9)(4,14,6,12)(10,16,13,15)(17,21,22,24)(18,25,26,19)(20,29,28,31)(23,32,30,27)(33,39,40,36)(34,44,45,42)(35,46,47,37)(38,43,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620025);