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 | |
4155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,10,14,8)(4,6,5,9)(12,13,15,16)(17,18)(19,22)(20,29)(21,25)(23,27)(24,26)(28,32)(30,31)(33,34,39,44)(35,40,46,48)(36,37,42,41)(38,47,45,43);(1,3)(2,8)(4,15)(5,12)(6,13)(7,14)(9,16)(10,11)(17,19)(18,24)(20,28)(21,31)(22,27)(23,26)(25,32)(29,30)(33,35,39,46)(34,40,44,48)(36,45,42,38)(37,43,41,47);(1,4,7,5)(2,9,11,6)(3,12,14,15)(8,13,10,16)(17,20,23,21)(18,25,27,29)(19,28,26,31)(22,30,24,32)(33,36)(34,41)(35,45)(37,44)(38,46)(39,42)(40,47)(43,48);(1,5,7,4)(2,6,11,9)(3,12,14,15)(8,13,10,16)(17,21,23,20)(18,25,27,29)(19,28,26,31)(22,32,24,30)(33,36)(34,37)(35,45)(38,46)(39,42)(40,43)(41,44)(47,48);(1,3)(2,10)(4,12)(5,15)(6,13)(7,14)(8,11)(9,16)(17,19)(18,22)(20,28)(21,31)(23,26)(24,27)(25,30)(29,32)(33,37)(34,42)(35,43)(36,44)(38,40)(39,41)(45,48)(46,47);(1,6)(2,4)(3,13)(5,11)(7,9)(8,15)(10,12)(14,16)(17,22)(18,26)(19,27)(20,30)(21,32)(23,24)(25,31)(28,29)(33,38,39,45)(34,43,44,47)(35,36,46,42)(37,40,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365115);