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 | |
4080 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(20,28)(24,29)(26,32)(30,31)(33,34,39,37)(35,41,46,40)(36,43,38,42)(44,47,45,48);(3,9)(7,13)(10,15)(14,16)(18,22)(23,25)(24,29)(30,31)(33,35)(34,40)(36,44)(37,41)(38,45)(39,46)(42,47)(43,48);(2,6)(7,13)(8,11)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,41)(36,44)(37,40)(38,45)(39,46)(42,48)(43,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(33,36,39,38)(34,42,37,43)(35,44,46,45)(40,47,41,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,34)(35,41,46,40)(36,43,38,42)(44,48,45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,36)(34,43,37,42)(35,45,46,44)(40,48,41,47)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002201);