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 | |
4105 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)(3,10)(4,14)(5,8)(6,15)(7,9)(11,16)(12,13)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34)(35,40)(36,37)(38,41)(39,42)(43,46)(44,45)(47,48);(1,3,7,5)(2,8,9,10)(4,12,15,11)(6,13,14,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34,39,42)(35,40,37,36)(38,41,47,48)(43,46,45,44);(1,4)(2,6)(3,11)(5,12)(7,15)(8,16)(9,14)(10,13)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,4)(2,6)(3,12)(5,11)(7,15)(8,13)(9,14)(10,16)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,36)(34,35)(37,42)(38,46)(39,40)(41,45)(43,48)(44,47);(1,5,7,3)(2,8,9,10)(4,12,15,11)(6,16,14,13)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,46,48,44);(1,6)(2,4)(3,13)(5,16)(7,14)(8,12)(9,15)(10,11)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315337);