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 | |
6082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,5)(3,7,8,6)(11,15)(13,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,29)(27,30)(31,32);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(18,23)(19,25)(20,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(21,26)(24,28)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,21,23,26)(20,27,29,31)(24,30,28,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,31)(26,32);(17,21)(18,25)(19,23)(20,30)(22,26)(24,31)(27,28)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=gi,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.
| maximal subgroup of | #72235981, #72262018, #72262162, … |
| maximal normal subgroup of | #72235981, #72262018, #72262162, … |
| maximal quotient of | #72235995, #72235997, #72300536 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8117051);