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 | |
4054 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)(19,24)(20,23)(21,25)(22,28)(26,31)(27,29)(30,32)(33,34,40,37)(35,44,39,41)(36,42,38,43)(45,48,46,47);(3,9)(7,13)(10,15)(14,16)(17,19,23,25)(18,24,20,21)(22,27,31,32)(26,30,28,29)(33,35,40,39)(34,41,37,44)(36,45,38,46)(42,47,43,48);(2,6)(7,13)(8,11)(14,16)(18,20)(19,25)(26,28)(27,32)(33,36,40,38)(34,42,37,43)(35,45,39,46)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,18)(19,24,25,21)(22,28,31,26)(27,30,32,29)(33,37,40,34)(35,41,39,44)(36,43,38,42)(45,47,46,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,24)(18,25,20,19)(22,30,31,29)(26,27,28,32)(33,38,40,36)(34,43,37,42)(35,45,39,46)(41,47,44,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,39,40,35)(34,41,37,44)(36,46,38,45)(42,47,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854567);