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 | |
4210 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)(3,13,14,8)(4,5,11,10)(6,16,15,12)(17,18)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3)(2,8)(4,15)(5,16)(6,11)(7,14)(9,13)(10,12)(18,24)(21,25)(26,32)(29,30)(36,44)(40,47)(42,48)(45,46);(2,9)(3,14)(4,11)(12,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4,7,11)(2,10,9,5)(3,15,14,6)(8,12,13,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,5,7,10)(2,11,9,4)(3,12,14,16)(6,8,15,13)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,7,15)(2,12,9,16)(3,11,14,4)(5,8,10,13)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,e^2=g,f^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, 9817372);