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 | |
5017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,20)(17,19);(5,7)(6,11)(8,9)(10,15)(12,19)(13,20)(14,18)(16,17);(1,2,3,4)(5,8)(6,12)(7,9)(10,16)(11,13)(14,20)(15,17)(18,19);(5,9)(6,13)(7,8)(10,17)(11,12)(14,19)(15,16)(18,20);(7,15)(8,16)(11,18)(12,20)Polycyclic presentation:
2^9|c^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240656, #72251367, #72251370, … |
| maximal normal subgroup of | #72240656, #72251367, #72251370, … |
| maximal quotient of | #72249354, #72267131, #72267169 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349928);