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 | |
4076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,11,12,6)(4,13,16,9)(5,15,14,10)(17,18)(19,21)(20,25)(22,27)(23,24)(26,31)(28,30)(29,32);(1,3,7,12)(2,6,8,11)(4,14,16,5)(9,10,13,15)(17,19)(18,21)(20,22)(23,26)(24,31)(25,27)(28,29)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(2,8)(6,11)(9,13)(10,15)(18,24)(19,26)(22,28)(25,32)(34,39)(37,47)(40,45)(41,48);(1,4,7,16)(2,9,8,13)(3,5,12,14)(6,15,11,10)(17,20)(18,25)(19,28)(21,30)(22,26)(23,29)(24,32)(27,31)(33,35)(34,40)(36,42)(37,41)(38,44)(39,45)(43,46)(47,48);(1,5)(2,10)(3,4)(6,13)(7,14)(8,15)(9,11)(12,16)(17,21,23,31)(18,26,24,19)(20,27,29,30)(22,25,28,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,48)(40,47)(42,44);(1,6,7,11)(2,3,8,12)(4,15,16,10)(5,13,14,9)(17,22,23,28)(18,27,24,30)(19,20,26,29)(21,25,31,32)(33,37,38,47)(34,42,39,43)(35,41,44,48)(36,45,46,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229384);