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 | |
4101 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)(3,13)(4,10)(5,15)(6,11)(7,9)(8,12)(14,16)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,22,28,27)(33,34,39,44)(35,40,45,48)(36,41,38,43)(37,42,47,46);(1,3,8,7)(2,9,12,13)(4,14,5,11)(6,15,16,10)(17,19)(18,24)(20,27)(21,26)(22,30)(23,31)(25,28)(29,32)(33,35)(34,40)(36,46)(37,41)(38,42)(39,45)(43,47)(44,48);(1,4,8,5)(2,10,12,15)(3,14,7,11)(6,13,16,9)(17,20)(18,25)(19,22)(21,24)(23,30)(26,29)(27,31)(28,32)(33,36,39,38)(34,41,44,43)(35,42,45,46)(37,40,47,48);(1,5,8,4)(2,10,12,15)(3,14,7,11)(6,9,16,13)(17,20)(18,26)(19,27)(21,24)(22,31)(23,30)(25,29)(28,32)(33,37)(34,42)(35,43)(36,40)(38,48)(39,47)(41,45)(44,46);(1,6,8,16)(2,11,12,14)(3,15,7,10)(4,13,5,9)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32)(33,35)(34,40)(36,42)(37,43)(38,46)(39,45)(41,47)(44,48);(1,7,8,3)(2,9,12,13)(4,11,5,14)(6,15,16,10)(17,22,23,27)(18,28,29,21)(19,30,31,20)(24,25,32,26)(33,38,39,36)(34,43,44,41)(35,42,45,46)(37,40,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8607773);