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 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(34,40)(36,44)(37,45)(41,48);(2,8)(3,4)(10,15)(11,12)(18,20)(21,26)(25,28)(29,32)(33,34)(35,41)(36,42)(37,46)(38,44)(39,40)(43,45)(47,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,35,39,47)(34,41,40,48)(36,45,44,37)(38,46,42,43);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,36)(34,42)(35,45)(37,47)(38,40)(39,44)(41,43)(46,48);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,37,39,45)(34,43,40,46)(35,36,47,44)(38,48,42,41);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,38,39,42)(34,44,40,36)(35,46,47,43)(37,48,45,41)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400134);