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 | |
3965 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)(4,8)(6,9)(7,10)(11,13)(12,15)(14,16)(17,18,23,25)(19,20,27,21)(22,30,32,24)(26,29,31,28)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(1,3,7,4)(2,8,10,5)(6,11,14,12)(9,15,16,13)(17,19)(18,20)(21,25)(22,31)(23,27)(24,29)(26,32)(28,30)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,46)(45,47);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(17,20,23,21)(18,19,25,27)(22,28,32,29)(24,26,30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,4,7,3)(2,5,10,8)(6,11,14,12)(9,15,16,13)(17,21,23,20)(18,19,25,27)(22,29,32,28)(24,26,30,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,5)(2,4)(3,10)(6,13)(7,8)(9,12)(11,16)(14,15)(17,22)(18,24)(19,26)(20,28)(21,29)(23,32)(25,30)(27,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,6,7,14)(2,9,10,16)(3,11,4,12)(5,13,8,15)(17,19)(18,21)(20,25)(22,26)(23,27)(24,29)(28,30)(31,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,e^2=hi,f^2=ghi,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, 10358844);