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 | |
9700 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)(6,9)(10,11)(13,14)(15,19)(16,17)(18,20);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20);(2,4)(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,19,20,18);(1,2)(3,4)(13,15)(14,19)(16,18)(17,20);(5,7)(6,10)(8,12)(9,11);(5,6)(7,10)(8,9)(11,12)Polycyclic presentation:
2^9|c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72232067, #72233758, #72234463, … |
| maximal normal subgroup of | #72232067, #72233758, #72234463, … |
| maximal quotient of | #72232144, #72281114, #72281356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581846);