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 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,14)(10,15)(11,16)(17,18)(19,25)(20,29)(21,24)(22,31)(23,26)(27,28)(30,32)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19,23,21)(18,24,26,25)(20,27,31,30)(22,28,29,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(2,7)(3,9)(8,14)(10,15)(17,19,23,21)(18,25,26,24)(20,30,31,27)(22,28,29,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3,6,9)(2,4,7,12)(5,10,13,15)(8,11,14,16)(17,20)(18,22)(19,27)(21,30)(23,31)(24,28)(25,32)(26,29)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,21,23,19)(18,24,26,25)(20,27,31,30)(22,32,29,28)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,30)(26,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=hi,e^2=gh,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, 8067084);