Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3655 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 3655 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,16,11)(8,9)(10,12)(14,15,20,19)(17,18);(5,7)(6,11)(8,14)(9,15)(10,16)(12,13)(17,20)(18,19);(2,4)(5,8)(6,9)(7,14)(10,17)(11,19)(12,18)(13,15)(16,20);(5,9)(6,8)(7,15)(10,18)(11,20)(12,17)(13,14)(16,19)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226647, #72227660, #72227678, … |
| maximal normal subgroup of | #72226647, #72227660, #72227678, … |
| maximal quotient of | #72219165, #72226638, #72233061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356665);