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 | |
3981 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34)(35,39)(36,37)(38,40)(41,45)(42,43)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,35,38,44)(34,39,40,46)(36,42,41,47)(37,43,45,48);(2,6)(7,14)(8,11)(15,16)(34,40)(36,41)(39,46)(42,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,36,38,41)(34,37,40,45)(35,42,44,47)(39,43,46,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,29)(18,26,22,30)(19,27,20,28)(24,31,25,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,37)(34,41)(35,43)(36,40)(38,45)(39,47)(42,46)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9771060);