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 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(18,21)(20,24)(25,29)(28,30)(33,34,40,38)(35,44,46,41)(36,47,42,37)(39,43,48,45);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,18,23,21)(19,24,27,20)(22,25,31,29)(26,30,32,28)(33,35,40,46)(34,41,38,44)(36,39,42,48)(37,43,47,45);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,23,27)(18,24,21,20)(22,26,31,32)(25,30,29,28)(33,36)(34,37)(35,39)(38,47)(40,42)(41,43)(44,45)(46,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,19)(21,27)(22,30)(23,24)(25,32)(26,29)(28,31)(33,37,40,47)(34,42,38,36)(35,45,46,43)(39,44,48,41);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,18)(19,20,27,24)(22,29,31,25)(26,28,32,30)(33,38,40,34)(35,44,46,41)(36,47,42,37)(39,45,48,43);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22,23,31)(18,25,21,29)(19,26,27,32)(20,28,24,30)(33,39,40,48)(34,43,38,45)(35,42,46,36)(37,41,47,44)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8337659);