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 | |
4029 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)(3,7)(4,10)(5,13)(6,8)(9,12)(11,16)(14,15)(17,18,22,24)(19,20,25,27)(21,23,30,32)(26,28,29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3,6,12)(2,7,8,9)(4,13,15,11)(5,14,16,10)(19,25)(20,27)(26,29)(28,31)(33,34,38,42)(35,39,40,36)(37,41,46,45)(43,47,48,44);(2,8)(3,12)(10,14)(11,13)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,30)(31,32)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,3)(2,9)(4,13)(5,14)(6,12)(7,8)(10,16)(11,15)(17,20)(18,19)(21,28)(22,27)(23,26)(24,25)(29,32)(30,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,4)(2,10)(3,13)(5,7)(6,15)(8,14)(9,16)(11,12)(17,19)(18,20)(21,29)(22,25)(23,31)(24,27)(26,30)(28,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,44)(47,48);(1,5,6,16)(2,11,8,13)(3,14,12,10)(4,9,15,7)(17,21)(18,23)(19,26)(20,28)(22,30)(24,32)(25,29)(27,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9844960);