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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(19,22)(24,26)(27,29)(31,32)(33,34)(35,41)(36,38)(37,44)(39,43)(40,45)(42,46)(47,48);(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,e^2=hi,f^2=gh,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, 8780792);