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 | |
5451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,24,27,29)(20,25,30,32)(21,26,31,28);(10,14)(11,15)(18,23)(19,27)(25,32)(26,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16);(3,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32);(1,3)(2,6)(4,7)(5,8)(17,20)(18,25)(19,28)(21,24)(22,30)(23,32)(26,27)(29,31);(1,2)(3,6)(4,5)(7,8)(17,21)(18,26)(19,25)(20,29)(22,31)(23,28)(24,30)(27,32)Polycyclic presentation:
2^9|a^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72326547 |
| maximal normal subgroup of | #72326547 |
| maximal quotient of | #72326590, #72326705, #72327454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358862);