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 | #51817 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(18,23)(19,25)(21,29)(27,32)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,7)(5,8)(9,14)(13,15)(17,18,22,23)(19,20,25,28)(21,30,29,24)(26,32,31,27)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(4,12)(9,14)(10,16)(13,15)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9151022);