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 | |
3954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(4,8)(9,15)(12,13)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,32,28,25)(33,34,39,43)(35,40,46,48)(36,37,45,41)(38,44,47,42);(1,2,6,7)(3,8,11,4)(5,9,14,15)(10,13,16,12)(17,18)(19,23)(20,24)(21,32)(22,26)(25,28)(27,30)(29,31)(33,35)(34,40)(36,47)(37,42)(38,45)(39,46)(41,44)(43,48);(1,3,6,11)(2,8,7,4)(5,10,14,16)(9,13,15,12)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,27)(31,32)(33,36,39,45)(34,41,43,37)(35,38,46,47)(40,42,48,44);(1,4)(2,3)(5,13)(6,8)(7,11)(9,16)(10,15)(12,14)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,25,27,32)(33,35)(34,40)(36,38)(37,44)(39,46)(41,42)(43,48)(45,47);(1,2)(3,8)(4,11)(5,9)(6,7)(10,13)(12,16)(14,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,24,27,31)(33,37)(34,36)(35,44)(38,40)(39,41)(42,46)(43,45)(47,48);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,18)(19,23)(20,31)(21,25)(22,26)(24,29)(27,30)(28,32)(33,38)(34,42)(35,45)(36,46)(37,40)(39,47)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358860);