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 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(8,14)(11,15)(33,34,39,43)(35,44,47,40)(36,38,46,41)(37,42,45,48);(1,2)(3,10)(4,9)(5,8)(6,7)(11,16)(12,15)(13,14)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35)(34,40)(36,45)(37,46)(38,42)(39,47)(41,48)(43,44);(3,9)(4,10)(5,13)(8,14)(18,22)(19,26)(20,29)(31,32)(33,36,39,46)(34,41,43,38)(35,45,47,37)(40,48,44,42);(1,3,6,9)(2,4,7,10)(5,11,13,15)(8,12,14,16)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36)(34,41)(35,37)(38,43)(39,46)(40,42)(44,48)(45,47);(1,4)(2,3)(5,12)(6,10)(7,9)(8,11)(13,16)(14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37)(34,42)(35,46)(36,47)(38,40)(39,45)(41,44)(43,48);(1,5,6,13)(2,8,7,14)(3,11,9,15)(4,12,10,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,41)(34,36,43,46)(35,42,47,48)(37,40,45,44)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965785);