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 | |
4116 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)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,39,44)(35,38,43,40)(36,41,37,42)(45,46,47,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35,39,43)(34,40,44,38)(36,45,37,47)(41,48,42,46);(2,6)(7,14)(8,11)(15,16)(18,24)(20,25)(23,30)(26,31)(33,36,39,37)(34,41,44,42)(35,45,43,47)(38,46,40,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,36)(34,42,44,41)(35,45,43,47)(38,46,40,48);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,38,39,40)(34,43,44,35)(36,46,37,48)(41,47,42,45)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=hi,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, 9771056);