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 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,29,30,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(2,6)(7,9)(8,12)(14,15)(20,28)(23,30)(27,31)(29,32)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,30)(26,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,22,26)(18,24,25,19)(20,29,28,32)(23,31,30,27)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,30)(31,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877290);