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 | |
4158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32)(33,34)(35,36)(37,45)(38,39)(40,47)(41,44)(42,48)(43,46);(2,6)(4,12)(7,10)(11,15)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,38,41)(34,36,39,44)(37,46,47,42)(40,48,45,43);(3,9)(7,10)(11,15)(14,16)(18,23)(22,25)(24,28)(31,32)(34,39)(35,41)(40,45)(42,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,47)(46,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,37)(34,40)(35,42)(36,43)(38,47)(39,45)(41,46)(44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,f^2=g,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, 10231826);