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 | |
4057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,15)(4,9,14,13)(5,12,16,10)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34,38,40)(35,39,42,36)(37,45,47,41)(43,44,48,46);(2,7)(4,14)(8,15)(10,12)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,38,42)(34,39,40,36)(37,43,47,48)(41,46,45,44);(2,7)(3,11)(9,13)(10,12)(19,26)(22,23)(27,32)(30,31)(34,40)(35,42)(41,45)(43,48);(1,3,6,11)(2,8,7,15)(4,10,14,12)(5,13,16,9)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,34)(35,39)(36,42)(37,41)(38,40)(43,46)(44,48)(45,47);(1,4)(2,9)(3,12)(5,15)(6,14)(7,13)(8,16)(10,11)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,36,38,39)(34,35,40,42)(37,46,47,44)(41,48,45,43);(1,5,6,16)(2,10,7,12)(3,13,11,9)(4,8,14,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37)(34,41)(35,43)(36,44)(38,47)(39,46)(40,45)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,e^2=gh,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, 9074884);