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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(35,44)(37,47)(39,40)(42,45);(2,4)(7,9)(8,12)(14,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,34,38,43)(35,40,44,39)(36,41,46,48)(37,42,47,45);(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,35,38,44)(34,39,43,40)(36,45,46,42)(37,41,47,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,40)(36,45)(37,41)(38,44)(39,43)(42,46)(47,48);(1,4,6,2)(3,9,11,7)(5,8,13,12)(10,14,16,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36)(34,41)(35,45)(37,40)(38,46)(39,47)(42,44)(43,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^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, 9817395);