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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,21)(20,28)(22,30)(23,24)(25,32)(26,27)(29,31)(33,34,39,43)(35,44,46,37)(36,41,48,40)(38,47,45,42);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19)(18,21)(20,29)(22,27)(23,25)(24,32)(26,30)(28,31)(33,35)(34,37)(36,38)(39,46)(40,47)(41,42)(43,44)(45,48);(2,8)(4,9)(7,14)(10,15)(18,24)(19,25)(20,30)(27,31)(33,36,39,48)(34,40,43,41)(35,38,46,45)(37,47,44,42);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20)(18,22)(19,26)(21,31)(23,30)(24,28)(25,29)(27,32)(33,36)(34,41)(35,45)(37,47)(38,46)(39,48)(40,43)(42,44);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21)(18,19)(20,31)(22,26)(23,32)(24,25)(27,30)(28,29)(33,37)(34,35)(36,47)(38,40)(39,44)(41,45)(42,48)(43,46);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22)(18,20)(19,27)(21,29)(23,28)(24,30)(25,31)(26,32)(33,38,39,45)(34,42,43,47)(35,36,46,48)(37,41,44,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854457);