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 | |
4153 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)(3,10)(9,14)(11,16)(33,34)(35,39)(36,40)(37,43)(38,41)(42,47)(44,46)(45,48);(1,2,6,7)(3,4,10,8)(5,9,13,14)(11,12,16,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,38,43)(34,39,41,37)(36,42,46,48)(40,47,44,45);(3,10)(4,8)(5,13)(9,14)(18,22)(19,26)(20,29)(31,32)(33,34,38,41)(35,37,43,39)(36,44,46,40)(42,47,48,45);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,34)(35,37)(36,44)(38,41)(39,43)(40,46)(42,47)(45,48);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=g,e^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, 8965471);