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 | |
4111 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)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,30,32)(33,34,39,37)(35,43,38,40)(36,42,46,41)(44,47,45,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,30)(31,32)(33,35,39,38)(34,40,37,43)(36,44,46,45)(41,47,42,48);(2,4)(3,9)(10,12)(11,16)(17,19,22,29)(18,23,26,24)(20,27,25,30)(21,28,31,32)(33,36,39,46)(34,41,37,42)(35,44,38,45)(40,47,43,48);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,48)(43,47);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,20)(18,21)(19,27)(22,25)(23,28)(24,32)(26,31)(29,30)(33,37,39,34)(35,40,38,43)(36,42,46,41)(44,47,45,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,30)(24,27)(29,32)(33,38,39,35)(34,43,37,40)(36,45,46,44)(41,48,42,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091907);