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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3833 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)(7,14)(10,12)(15,16)(17,18,22,23)(19,24,28,32)(20,21,25,31)(26,30,29,27)(33,34,39,43)(35,37,41,48)(36,42,38,40)(44,47,45,46);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,23)(20,25)(24,32)(26,29)(33,35)(34,37)(36,44)(38,45)(39,41)(40,46)(42,47)(43,48);(2,6)(3,9)(8,11)(10,12)(17,19,22,28)(18,24,23,32)(20,26,25,29)(21,30,31,27)(33,36,39,38)(34,40,43,42)(35,44,41,45)(37,46,48,47);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,32)(25,26)(30,31);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,22,25)(18,21,23,31)(19,26,28,29)(24,30,32,27)(33,37)(34,41)(35,43)(36,46)(38,47)(39,48)(40,45)(42,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21)(18,25)(19,27)(20,23)(22,31)(24,26)(28,30)(29,32)(33,38,39,36)(34,42,43,40)(35,45,41,44)(37,47,48,46)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=i,e^2=g,f^2=h,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, 8837624);