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 | |
4259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(34,38)(35,41)(40,43)(42,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(2,9)(3,4)(10,16)(11,12)(18,23)(22,29)(24,27)(30,31)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,20)(18,24)(19,25)(21,28)(22,30)(23,27)(26,32)(29,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9782099);