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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,10)(11,16)(14,15)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(34,40)(36,41)(42,46)(44,48);(2,3)(7,9)(8,10)(14,15)(20,29)(24,28)(27,31)(30,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,23,29)(18,24,21,28)(19,27,25,31)(22,30,26,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,26,25,22)(20,24,29,28)(27,30,31,32)(33,37,39,35)(34,36,40,41)(38,45,47,43)(42,44,46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=gi,e^2=gi,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074634);