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 | |
4114 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)(3,7)(4,10)(5,13)(6,8)(9,12)(11,16)(14,15)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34)(35,38)(36,37)(39,43)(40,42)(41,47)(44,48)(45,46);(1,3,6,12)(2,7,8,9)(4,13,15,11)(5,14,16,10)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35)(34,40)(36,44)(37,46)(38,43)(39,42)(41,48)(45,47);(2,8)(3,12)(10,14)(11,13)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,39,47)(34,41,43,37)(35,44,42,45)(38,46,40,48);(1,3)(2,9)(4,13)(5,14)(6,12)(7,8)(10,16)(11,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36)(34,37)(35,45)(38,46)(39,47)(40,48)(41,43)(42,44);(1,4)(2,10)(3,13)(5,7)(6,15)(8,14)(9,16)(11,12)(17,20)(18,23)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28)(33,37)(34,36)(35,46)(38,45)(39,41)(40,44)(42,48)(43,47);(1,5,6,16)(2,11,8,13)(3,14,12,10)(4,9,15,7)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38)(34,42)(35,43)(36,46)(37,44)(39,40)(41,45)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,f^2=gh,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, 9844982);