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 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(19,25)(22,26)(27,31)(30,32)(33,34,39,42)(35,41,37,40)(36,45,47,38)(43,44,46,48);(1,2,6,3)(4,7,12,9)(5,10,13,8)(11,15,16,14)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,35,39,37)(34,40,42,41)(36,46,47,43)(38,48,45,44);(2,3)(7,9)(8,10)(14,15)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,36)(34,38)(35,43)(37,46)(39,47)(40,44)(41,48)(42,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,37,39,35)(34,41,42,40)(36,46,47,43)(38,48,45,44);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664465);