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 | |
4107 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)(19,25)(20,28)(21,27)(22,23)(24,29)(26,30)(31,32)(33,34,38,42)(35,36,40,39)(37,41,46,48)(43,44,45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(18,23)(20,29)(21,25)(26,32)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,30,28,31)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,28)(25,31)(27,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,48)(45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,21,22,25)(18,19,23,27)(20,30,29,31)(24,26,28,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10247424);