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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3904 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,14)(15,16)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30)(34,39)(35,42)(41,44)(43,48);(2,7)(6,9)(8,12)(14,15)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,13)(15,16)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103575);