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 |
| prime radical | #2543 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34)(35,38)(36,37)(39,42)(40,46)(41,47)(43,48)(44,45);(2,6)(7,9)(8,12)(14,15)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,41)(34,38,42,47)(36,44,46,43)(37,45,40,48);(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,39,46)(34,40,42,37)(35,43,41,44)(38,45,47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,36)(35,45)(38,44)(39,40)(41,48)(42,46)(43,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,38)(34,41)(35,42)(36,45)(37,43)(39,47)(40,44)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=g,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, 9690194);