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 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,16)(4,14,12,6)(5,9,15,13)(17,18,23,25)(19,24,27,32)(20,28,29,26)(21,22,30,31)(33,34)(35,41)(36,40)(37,43)(38,39)(42,47)(44,46)(45,48);(1,3)(2,8)(4,5)(6,13)(7,11)(9,14)(10,16)(12,15)(17,19)(18,24)(20,21)(22,28)(23,27)(25,32)(26,31)(29,30)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,47)(46,48);(3,11)(4,12)(6,14)(8,16)(18,25)(19,27)(20,29)(22,31)(34,39)(37,41)(40,44)(45,47);(1,4)(2,6)(3,5)(7,12)(8,13)(9,16)(10,14)(11,15)(17,20,23,29)(18,26,25,28)(19,21,27,30)(22,24,31,32)(33,34)(35,37)(36,44)(38,39)(40,46)(41,43)(42,47)(45,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,13)(14,16)(17,21)(18,22)(19,20)(23,30)(24,28)(25,31)(26,32)(27,29)(33,36,38,46)(34,40,39,44)(35,42,43,48)(37,45,41,47);(1,6)(2,4)(3,13)(5,8)(7,14)(9,11)(10,12)(15,16)(17,22)(18,21)(19,28)(20,24)(23,31)(25,30)(26,27)(29,32)(33,37)(34,35)(36,45)(38,41)(39,43)(40,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835961);