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 | #57158 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3956 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,16)(13,15)(17,18,22,25)(19,23,27,20)(21,24,30,32)(26,31,29,28)(34,39)(35,37)(41,44)(43,45);(1,3,5,11)(2,6,9,7)(4,13,14,10)(8,16,12,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,34,38,39)(35,40,37,42)(36,44,46,41)(43,48,45,47);(1,2,5,9)(3,7,11,6)(4,8,14,12)(10,16,13,15)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,16)(12,15)(13,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4)(2,8)(3,10)(5,14)(6,15)(7,16)(9,12)(11,13)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757817);