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 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,23,26)(19,22,21,25)(20,29,31,24)(27,30,28,32)(33,34,39,37)(35,38,44,40)(36,42,46,41)(43,45,48,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,23,21)(18,22,26,25)(20,28,31,27)(24,30,29,32)(33,35,39,44)(34,40,37,38)(36,43,46,48)(41,45,42,47);(2,6)(3,9)(8,11)(10,12)(17,20,23,31)(18,24,26,29)(19,27,21,28)(22,32,25,30)(33,36,39,46)(34,41,37,42)(35,43,44,48)(38,47,40,45);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20)(18,24)(19,28)(21,27)(22,30)(23,31)(25,32)(26,29)(33,36)(34,42)(35,43)(37,41)(38,45)(39,46)(40,47)(44,48);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21,23,19)(18,25,26,22)(20,28,31,27)(24,30,29,32)(33,37,39,34)(35,38,44,40)(36,42,46,41)(43,47,48,45);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22,23,25)(18,19,26,21)(20,30,31,32)(24,28,29,27)(33,38)(34,35)(36,45)(37,44)(39,40)(41,48)(42,43)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838651);