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 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,12,13,8)(10,14,16,15)(17,18)(19,20)(21,29)(22,23)(24,26)(25,30)(27,31)(28,32)(37,45)(40,47)(43,44)(46,48);(1,3,5,11)(2,6,7,9)(4,10,13,16)(8,14,12,15)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,7)(6,9)(8,12)(14,15)(18,23)(19,26)(25,29)(27,32)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025995);