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 | |
4257 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)(3,7)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(17,18)(19,21)(20,23)(22,28)(24,25)(26,30)(27,31)(29,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,19,23,25)(18,24,20,21)(22,27,30,32)(26,31,28,29)(34,35)(39,41)(40,42)(46,47);(2,8)(3,9)(4,13)(11,16)(18,20)(21,24)(26,28)(29,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,24,25,21)(22,28,30,26)(27,31,32,29)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,21,23,24)(18,25,20,19)(22,29,30,31)(26,32,28,27)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,22,23,30)(18,26,20,28)(19,27,25,32)(21,29,24,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034460);