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 | |
4260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,10)(8,12)(14,16)(17,18,22,20)(19,23,25,27)(21,24,30,28)(26,31,29,32)(33,34,39,43)(35,42,46,40)(36,38,37,41)(44,45,47,48);(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35)(34,40)(36,47)(37,44)(38,45)(39,46)(41,48)(42,43);(3,9)(7,10)(11,15)(14,16)(19,25)(23,27)(26,29)(31,32)(33,36,39,37)(34,41,43,38)(35,44,46,47)(40,45,42,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,37,39,36)(34,38,43,41)(35,44,46,47)(40,45,42,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,35)(34,42)(36,44)(37,47)(38,45)(39,46)(40,43)(41,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,38)(34,36)(35,45)(37,43)(39,41)(40,47)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259623);