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 | |
4099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,15)(11,16)(13,14)(17,18,22,26)(19,23,28,25)(20,29,21,24)(27,32,30,31)(33,34,40,39)(35,45,47,41)(36,42,46,48)(37,43,38,44);(1,2)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(17,20,22,21)(18,24,26,29)(19,27,28,30)(23,31,25,32)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,4,6,12)(2,7,10,15)(3,11,9,16)(5,14,8,13)(17,18)(19,25)(20,24)(21,29)(22,26)(23,28)(27,32)(30,31)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,19)(18,25)(20,27)(21,30)(22,28)(23,26)(24,32)(29,31)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,21,22,20)(18,24,26,29)(19,27,28,30)(23,32,25,31)(33,39,40,34)(35,45,47,41)(36,48,46,42)(37,43,38,44)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595563);