Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3070 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 3070 conjugacy classes of subgroups — too many to draw. Only its 346 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,19)(14,16)(15,18)(17,20);(10,16)(17,19);(1,3,4,2)(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,20)(18,19);(5,8)(6,11)(7,13)(9,15)(10,17)(12,18)(14,20)(16,19)Polycyclic presentation:
2^9|a^2=h,c^2=h,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225619, #72225639, #72225646, … |
| maximal normal subgroup of | #72225619, #72225639, #72225646, … |
| maximal quotient of | #72242984, #72247316, #72251581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280833);