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 | |
4090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,20,27,23)(21,25,30,32)(26,28,29,31)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,32)(29,30)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(2,7)(3,10)(8,15)(11,12)(18,24)(20,23)(25,32)(28,31)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,21,22,30)(18,25,24,32)(19,26,27,29)(20,28,23,31)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770588);