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 | |
3954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,20)(19,25,21,24)(22,28,30,26)(27,31,29,32)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(19,21)(22,30)(24,25)(26,28)(34,39)(35,41)(42,45)(43,48);(2,6)(7,10)(8,12)(14,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8732425);