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 | |
4314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,15)(4,8,13,16)(10,14,11,12)(17,18,23,25)(19,26,21,24)(20,29,31,22)(27,28,30,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19,23,21)(18,24,25,26)(20,27,31,30)(22,28,29,32)(35,41)(38,45)(42,47)(46,48);(2,7)(3,9)(8,16)(10,11)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3)(2,5)(4,10)(6,9)(7,15)(8,14)(11,13)(12,16)(17,20)(18,22)(19,27)(21,30)(23,31)(24,28)(25,29)(26,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,21,23,19)(18,24,25,26)(20,30,31,27)(22,28,29,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,31)(26,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592795);