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 | |
4093 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)(19,25,29,23)(20,30,26,21)(27,28,32,31)(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)(20,27)(21,28)(22,29)(24,25)(26,32)(30,31)(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)(20,26)(23,25)(27,32)(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)(17,20,22,26)(18,21,24,30)(19,27,29,32)(23,28,25,31)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,19)(18,25)(20,27)(21,31)(22,29)(23,24)(26,32)(28,30)(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)(17,21)(18,26)(19,28)(20,24)(22,30)(23,32)(25,27)(29,31)(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=ghi,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, 9770600);