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 | |
4022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,7)(3,12,14,9)(4,10,5,15)(6,11,13,16)(17,18,23,26)(19,22,24,20)(21,25,30,32)(27,31,29,28)(34,39)(36,45)(40,46)(43,48);(1,3,8,14)(2,9,7,12)(4,13,5,6)(10,11,15,16)(17,19)(18,22)(20,26)(21,29)(23,24)(25,28)(27,30)(31,32)(35,42)(40,46)(43,48)(44,47);(1,4,8,5)(2,10,7,15)(3,13,14,6)(9,11,12,16)(17,20,23,22)(18,24,26,19)(21,28,30,31)(25,29,32,27)(33,34,38,39)(35,40,42,46)(36,41,45,37)(43,47,48,44);(1,5,8,4)(2,10,7,15)(3,6,14,13)(9,11,12,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,29)(26,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,46)(41,47)(45,48);(1,6)(2,11)(3,4)(5,14)(7,16)(8,13)(9,15)(10,12)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(33,36)(34,41)(35,43)(37,39)(38,45)(40,47)(42,48)(44,46);(1,7,8,2)(3,12,14,9)(4,15,5,10)(6,11,13,16)(17,22,23,20)(18,24,26,19)(21,28,30,31)(25,27,32,29)(33,37,38,41)(34,36,39,45)(35,44,42,47)(40,43,46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8570120);