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 | |
4100 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)(4,14)(5,11)(6,15)(7,9)(10,13)(12,16)(17,18,23,24)(19,25,28,21)(20,22,30,32)(26,27,29,31)(34,39)(35,42)(37,45)(44,47);(1,3,7,12)(2,8,9,16)(4,6,13,11)(5,14,15,10)(17,19,23,28)(18,21,24,25)(20,29,30,26)(22,27,32,31)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,9)(3,12)(4,13)(5,15)(18,24)(20,30)(21,25)(26,29)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4,7,13)(2,10,9,14)(3,11,12,6)(5,8,15,16)(17,20)(18,22)(19,26)(21,31)(23,30)(24,32)(25,27)(28,29)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5,7,15)(2,11,9,6)(3,10,12,14)(4,8,13,16)(17,21)(18,19)(20,31)(22,26)(23,25)(24,28)(27,30)(29,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,6,7,11)(2,5,9,15)(3,13,12,4)(8,14,16,10)(17,22)(18,20)(19,27)(21,29)(23,32)(24,30)(25,26)(28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9943906);