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 | |
4051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,10)(11,16)(14,15)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(2,3)(7,9)(8,10)(14,15)(20,29)(24,28)(27,31)(30,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,26,25,22)(20,24,29,28)(27,30,31,32)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=gi,f^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, 8074626);