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 | |
4058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,2)(3,6)(4,8)(5,13)(7,10)(9,15)(11,16)(12,14)(19,25)(22,28)(26,31)(29,32)(34,38)(39,45)(40,43)(46,47);(1,3,7,4)(2,8,10,6)(5,11,15,12)(9,16,13,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,7,3)(2,8,10,6)(5,12,15,11)(9,16,13,14)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,40)(35,41)(37,44)(38,43)(39,46)(42,48)(45,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(1,6,7,8)(2,3,10,4)(5,14,15,16)(9,11,13,12)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,35)(34,39)(36,41)(37,42)(38,45)(40,46)(43,47)(44,48)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8154869);