Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437632 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(10,16)(11,18);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(5,8)(6,11)(7,13)(9,15)(10,17)(12,18)(14,19)(16,20);(7,14)(10,16)(13,19)(17,20)Polycyclic presentation:
2^9|c^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225623, #72225633, #72225635, … |
| maximal normal subgroup of | #72225623, #72225633, #72225635, … |
| maximal quotient of | #72240570, #72249354, C2×#6442728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349924);