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 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,5,9,7)(4,8,14,16)(11,13,12,15)(20,28)(24,29)(26,32)(30,31)(33,34,38,43)(35,42,46,39)(36,40,41,37)(44,47,48,45);(1,3,6,9)(2,5,10,7)(4,12,14,11)(8,15,16,13)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35)(34,39)(36,44)(37,47)(38,46)(40,45)(41,48)(42,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36,38,41)(34,40,43,37)(35,44,46,48)(39,45,42,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,45)(47,48);(1,4,6,14)(2,8,10,16)(3,11,9,12)(5,13,7,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,37,38,40)(34,41,43,36)(35,45,46,47)(39,44,42,48);(1,5)(2,9)(3,10)(4,13)(6,7)(8,12)(11,16)(14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,35)(34,42)(36,44)(37,45)(38,46)(39,43)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379612);