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 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,24,27)(19,28,30,25)(20,22,21,31)(23,32,29,26)(35,42)(37,45)(39,40)(41,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,23)(21,29)(22,26)(24,30)(27,28)(31,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,20,24,21)(18,22,27,31)(19,23,30,29)(25,26,28,32)(33,35)(34,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,21,24,20)(18,22,27,31)(19,29,30,23)(25,26,28,32)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,22)(18,21)(19,26)(20,27)(23,28)(24,31)(25,29)(30,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,23,24,29)(18,26,27,32)(19,20,30,21)(22,28,31,25)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044830);