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 | |
4090 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(18,22)(20,27)(23,29)(25,28)(33,34)(35,44)(36,37)(38,48)(39,43)(40,46)(41,47)(42,45);(2,7)(3,10)(4,12)(13,14)(18,22)(23,29)(24,30)(31,32)(33,35)(34,40)(36,38)(37,42)(39,46)(41,48)(43,44)(45,47);(17,18,21,22)(19,23,26,29)(20,24,27,30)(25,31,28,32)(33,36,39,47)(34,41,43,37)(35,38,46,45)(40,48,44,42);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,13,15,14)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32)(33,36)(34,37)(35,45)(38,46)(39,47)(40,48)(41,43)(42,44);(1,4,6,12)(2,9,7,15)(3,11,10,16)(5,14,8,13)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31)(33,37)(34,36)(35,42)(38,40)(39,41)(43,47)(44,45)(46,48);(1,5,6,8)(2,10,7,3)(4,13,12,14)(9,11,15,16)(17,20)(18,24)(19,25)(21,27)(22,30)(23,31)(26,28)(29,32)(33,38,39,45)(34,42,43,48)(35,36,46,47)(37,44,41,40)Polycyclic presentation:
2^9|c^2=h,d^2=ghi,e^2=gh,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, 9852735);