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 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(33,34)(35,41)(36,45)(37,40)(38,47)(39,42)(43,44)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,37)(34,40,42,41)(36,43,47,46)(38,44,45,48);(2,6)(7,9)(8,13)(14,15)(18,20)(24,25)(26,28)(31,32)(33,35,39,37)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36)(34,38)(35,43)(37,46)(39,47)(40,44)(41,48)(42,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,39,35)(34,40,42,41)(36,43,47,46)(38,48,45,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062215);