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 | |
4264 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)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,21)(19,22,25,27)(20,24,29,30)(26,28,31,32)(33,34,40,43)(35,44,46,41)(36,38,37,47)(39,42,48,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,28)(29,31)(30,32)(33,35)(34,41)(36,42)(37,45)(38,39)(40,46)(43,44)(47,48);(2,6)(7,9)(8,12)(15,16)(19,25)(22,27)(26,31)(28,32)(33,36,40,37)(34,38,43,47)(35,42,46,45)(39,44,48,41);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,31)(27,32)(33,37,40,36)(34,38,43,47)(35,45,46,42)(39,44,48,41);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21,23,18)(19,27,25,22)(20,24,29,30)(26,28,31,32)(33,38)(34,37)(35,39)(36,43)(40,47)(41,45)(42,44)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,22,23,27)(18,25,21,19)(20,28,29,32)(24,31,30,26)(33,39)(34,42)(35,38)(36,41)(37,44)(40,48)(43,45)(46,47)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8248796);