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 | |
4338 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(19,25)(23,26)(27,31)(30,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(2,7)(4,8)(10,14)(12,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(3,9)(4,8)(5,13)(10,14)(18,22)(19,25)(20,28)(30,32)(33,34,38,41)(35,39,36,42)(37,45,46,40)(43,48,44,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957662);