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 | |
3953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30)(33,34)(35,40)(36,42)(37,41)(38,43)(39,44)(45,46)(47,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,37)(34,40,44,41)(36,47,43,45)(38,46,42,48);(3,11)(7,8)(12,16)(13,14)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,37)(34,41,44,40)(36,45,43,47)(38,46,42,48);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32)(33,36,39,43)(34,42,44,38)(35,45,37,47)(40,46,41,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,20)(18,19)(21,30)(22,23)(24,29)(25,27)(26,32)(28,31)(33,37,39,35)(34,40,44,41)(36,47,43,45)(38,48,42,46);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,32)(27,29)(33,38,39,42)(34,43,44,36)(35,46,37,48)(40,45,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641119);