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 | |
4136 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)(3,8,4,13)(5,10,15,14)(6,12,16,11)(19,25)(23,26)(27,31)(30,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,46)(44,48)(45,47);(1,3,7,4)(2,8,9,13)(5,12,15,11)(6,10,16,14)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(2,9)(3,4)(5,15)(6,16)(18,22)(19,25)(20,28)(30,32)(34,38)(36,42)(37,45)(43,48);(1,4,7,3)(2,8,9,13)(5,12,15,11)(6,14,16,10)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,39,42)(34,40,38,35)(37,46,45,44)(41,43,47,48);(1,5)(2,10)(3,12)(4,11)(6,13)(7,15)(8,16)(9,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37,39,45)(34,41,38,47)(35,43,40,48)(36,44,42,46);(1,6,7,16)(2,11,9,12)(3,10,4,14)(5,13,15,8)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,34)(35,42,40,36)(37,47,45,41)(43,46,48,44)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964516);