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 | |
4140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(4,12)(7,10)(8,13)(17,18,23,24)(19,26,29,20)(21,22,31,25)(27,30,32,28)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,47)(46,48);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,19,23,29)(18,20,24,26)(21,27,31,32)(22,28,25,30)(33,35,38,40)(34,39,42,36)(37,43,46,45)(41,47,48,44);(2,6)(3,9)(8,13)(11,14)(18,24)(20,26)(22,25)(28,30)(33,35,38,40)(34,36,42,39)(37,45,46,43)(41,47,48,44);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,20)(18,19)(21,28)(22,27)(23,26)(24,29)(25,32)(30,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,21)(18,25)(19,27)(20,30)(22,24)(23,31)(26,28)(29,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,45)(43,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22,23,25)(18,21,24,31)(19,28,29,30)(20,27,26,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840271);