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 | |
4041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,6)(7,9)(8,13)(14,15)(17,19,22,27)(18,20,25,23)(21,26,30,32)(24,28,29,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000785);