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 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,28)(29,32)(33,34)(35,40)(36,45)(37,47)(38,41)(39,42)(43,44)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(35,42)(39,40)(43,46)(44,48);(2,6)(7,9)(8,13)(14,15)(18,24)(20,25)(26,30)(28,32)(33,35,38,42)(34,39,41,40)(36,43,47,46)(37,44,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,35)(34,40)(36,46)(37,44)(38,42)(39,41)(43,47)(45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,36)(34,37)(35,43)(38,47)(39,44)(40,48)(41,45)(42,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,37)(34,36)(35,44)(38,45)(39,43)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8061991);