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 | |
4146 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)(17,18,23,20)(19,27,29,24)(21,25,28,30)(22,31,32,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(34,39)(40,41)(42,45)(47,48);(2,9)(3,4)(10,16)(11,12)(18,20)(24,27)(25,30)(26,31)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,23,18)(19,27,29,24)(21,30,28,25)(22,31,32,26)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817439);