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 | |
4039 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,21,23)(19,24,27,22)(20,28,30,25)(26,31,29,32)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,21,27)(18,22,23,24)(20,26,30,29)(25,31,28,32)(33,35)(34,41)(36,39)(37,46)(38,43)(40,47)(42,48)(44,45);(2,6)(7,9)(8,12)(15,16)(18,23)(22,24)(25,28)(31,32)(33,36,40,37)(34,42,44,38)(35,39,47,46)(41,48,45,43);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,22)(20,25)(21,23)(24,27)(26,31)(28,30)(29,32)(33,37,40,36)(34,42,44,38)(35,46,47,39)(41,48,45,43);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,19)(18,24)(20,29)(21,27)(22,23)(25,31)(26,30)(28,32)(33,38,40,42)(34,37,44,36)(35,43,47,48)(39,41,46,45);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,20)(18,25)(19,26)(21,30)(22,31)(23,28)(24,32)(27,29)(33,39,40,46)(34,43,44,48)(35,36,47,37)(38,45,42,41)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050978);