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 | |
4164 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,21)(19,24,22,29)(20,25,27,30)(26,31,28,32)(33,34,38,40)(35,41,43,37)(36,39,46,44)(42,47,48,45);(2,7)(4,15)(8,11)(10,13)(17,19,23,22)(18,24,21,29)(20,26,27,28)(25,31,30,32)(33,35)(34,37)(36,42)(38,43)(39,45)(40,41)(44,47)(46,48);(2,7)(3,12)(4,15)(5,16)(18,21)(19,22)(20,27)(31,32)(33,34,38,40)(35,37,43,41)(36,44,46,39)(42,47,48,45);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,25)(19,26)(21,30)(22,28)(23,27)(24,31)(29,32)(33,34)(35,37)(36,44)(38,40)(39,46)(41,43)(42,47)(45,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,18)(19,24,22,29)(20,25,27,30)(26,32,28,31)(33,36,38,46)(34,39,40,44)(35,42,43,48)(37,45,41,47);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,19)(18,24,21,29)(20,28,27,26)(25,31,30,32)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8923777);