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 | |
3946 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,40,43)(35,41,45,48)(36,46,38,42)(37,39,47,44);(2,8)(5,11)(9,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,40,45)(34,41,43,48)(36,44,38,39)(37,42,47,46);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,36,40,38)(34,42,43,46)(35,44,45,39)(37,41,47,48);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37)(34,39)(35,42)(36,41)(38,48)(40,47)(43,44)(45,46);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,38,40,36)(34,42,43,46)(35,39,45,44)(37,41,47,48);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,39)(34,37)(35,36)(38,45)(40,44)(41,42)(43,47)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358779);