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 | |
5389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(11,16)(12,14)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,30)(29,32);(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,15)(14,16)(17,19,21,27)(18,22,25,23)(20,28,29,26)(24,31,32,30);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,27)(18,23,25,22)(20,28,29,26)(24,30,32,31);(10,15)(11,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,31)(26,29)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,20,21,29)(18,24,25,32)(19,26,27,28)(22,30,23,31);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279754 |
| maximal normal subgroup of | #72279754 |
| maximal quotient of | #72280373, #72280982, #72281123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968287);