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 | |
5198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(18,19)(23,25)(24,26)(30,31);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(20,27)(21,29)(23,25)(24,26);(3,7)(6,8)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^9|b^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280742, #72300627 |
| maximal normal subgroup of | #72280742, #72300627 |
| maximal quotient of | #72281224, #72281931, #72300634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842134);