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 | |
4231 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,24)(19,23,26,20)(21,29,31,25)(27,28,32,30)(33,34)(35,45)(36,39)(37,42)(38,43)(40,44)(41,48)(46,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,29)(31,32)(33,35)(34,41)(36,38)(37,46)(39,47)(40,48)(42,43)(44,45);(2,6)(7,13)(8,11)(14,16)(18,24)(19,26)(25,29)(27,32)(33,36,40,37)(34,39,44,42)(35,38,48,46)(41,47,45,43);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(33,37,40,36)(34,42,44,39)(35,46,48,38)(41,43,45,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,19)(21,30)(22,23)(24,26)(25,32)(27,29)(28,31)(33,38,40,46)(34,43,44,47)(35,36,48,37)(39,41,42,45);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,31)(18,25,24,29)(19,27,26,32)(20,28,23,30)(33,39,40,42)(34,36,44,37)(35,47,48,43)(38,45,46,41)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9625715);