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 | |
4252 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)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(35,41)(39,42)(43,47)(46,48);(2,6)(7,13)(8,14)(15,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(3,9)(7,13)(10,11)(15,16)(18,25)(19,20)(26,32)(27,28)(34,38)(39,42)(40,44)(46,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9930326);