Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
9953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(13,14,17,16)(15,18,19,20);(6,9)(10,12)(14,16)(18,20);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20);(7,11)(10,12)(15,19)(18,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,19);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,20,19,18)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72232031, #72232070, #72233523, … |
| maximal normal subgroup of | #72232031, #72232070, #72233523, … |
| maximal quotient of | #72232038, #72232109, #72232159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538503);