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 | |
4302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(19,23)(20,27)(26,31)(28,32)(33,35,38,43)(34,39,41,36)(37,42,46,47)(40,45,48,44);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,36)(34,35)(37,45)(38,39)(40,47)(41,43)(42,48)(44,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,37,38,46)(34,40,41,48)(35,42,43,47)(36,44,39,45)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,f^2=g,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, 10330204);