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 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,10)(11,16)(14,15)(18,22)(23,28)(24,29)(30,32)(34,38)(35,41)(36,44)(46,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(19,25)(23,28)(26,31)(30,32)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|b^2=gi,d^2=i,e^2=i,f^2=h,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, 8074643);