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 | |
4091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,4)(5,13,14,9)(11,12,16,15)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(33,34,39,37)(35,41,38,40)(36,44,46,42)(43,47,45,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,15)(12,13)(14,16)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(33,35,39,38)(34,40,37,41)(36,43,46,45)(42,47,44,48);(2,8)(3,10)(9,13)(11,16)(19,26)(21,24)(27,32)(29,31)(33,35,39,38)(34,41,37,40)(36,43,46,45)(42,48,44,47);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,42)(35,43)(37,44)(38,45)(39,46)(40,47)(41,48);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,34)(35,41,38,40)(36,44,46,42)(43,48,45,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38,39,35)(34,41,37,40)(36,45,46,43)(42,48,44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032684);