Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
19350 conjugacy classes of subgroups; only the 1866 normal subgroups are shown.
This group has 19350 conjugacy classes of subgroups — too many to draw. Only its 1866 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,8)(13,14,18,17)(15,20,16,19);(10,12)(13,15,18,16)(14,19,17,20);(1,2,3,4)(5,6,7,8)(9,10,11,12);(13,16,18,15)(14,19,17,20);(13,17,18,14)(15,20,16,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72297967, #72297970, #72297986, … |
| maximal normal subgroup of | #72297967, #72297970, #72297986, … |
| maximal quotient of | #72298417, #72337339, #72337387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424813);