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 | |
9399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(7,11)(10,12)(13,14,17,16)(15,19,20,18);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19);(2,4)(6,9)(10,12);(1,2)(3,4)(13,14,17,16)(15,18,20,19);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72232032, #72234901, #72234902, … |
| maximal normal subgroup of | #72232032, #72234901, #72234902, … |
| maximal quotient of | #72232056, #72232338, #72260984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577420);