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 | |
4092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,12,14,9)(11,15,13,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34)(35,41)(36,44)(37,43)(38,42)(39,40)(45,48)(46,47);(1,3,7,5)(2,8,6,10)(4,11,14,13)(9,15,12,16)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,35,40,47)(34,41,39,46)(36,45,38,43)(37,44,48,42);(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,36,40,38)(34,42,39,44)(35,43,47,45)(37,41,48,46);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,37,40,48)(34,43,39,45)(35,44,47,42)(36,46,38,41);(1,5,7,3)(2,8,6,10)(4,13,14,11)(9,15,12,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,38,40,36)(34,42,39,44)(35,45,47,43)(37,41,48,46);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,39,40,34)(35,46,47,41)(36,44,38,42)(37,45,48,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9581472);