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 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34,40,42)(35,36,37,38)(39,47,48,41)(43,46,45,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,36)(37,40)(38,42)(39,43)(41,44)(45,48)(46,47);(2,6)(7,9)(8,12)(14,15)(18,20)(24,26)(25,28)(31,32)(33,36,40,38)(34,35,42,37)(39,44,48,46)(41,43,47,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,37,40,35)(34,36,42,38)(39,45,48,43)(41,44,47,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,38,40,36)(34,35,42,37)(39,44,48,46)(41,45,47,43);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,39)(34,41)(35,43)(36,44)(37,45)(38,46)(40,48)(42,47)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854672);