Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,14,16,10)(8,11)(9,13)(12,17)(15,18,20,19);(1,2,3,4)(5,7)(6,10)(8,15)(9,16)(11,19)(12,18)(13,14)(17,20);(2,4)(5,8)(6,11)(7,15)(9,17)(10,18)(12,13)(14,19)(16,20);(5,8)(6,12,13,11)(7,15)(9,17)(10,18,14,19)(16,20)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225630, #72225636, #72225642, … |
| maximal normal subgroup of | #72225630, #72225636, #72225642, … |
| maximal quotient of | #72251470, #72265828, #72265832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371936);