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 | |
4047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,14,16,15)(21,29)(24,26)(27,32)(30,31)(33,34,38,42)(35,39,36,40)(37,45,46,41)(43,48,44,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(2,6)(7,9)(8,12)(14,15)(18,19)(23,25)(24,26)(30,31)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,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, 8074630);