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 | |
4107 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)(3,9)(4,11)(5,13)(6,7)(8,15)(10,12)(14,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(2,7)(4,9)(8,13)(12,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,3)(2,4)(5,14)(6,11)(7,9)(8,16)(10,15)(12,13)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,5,6,15)(2,8,7,13)(3,10,11,14)(4,12,9,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gh,f^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, 10327380);