Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(5,6)(7,13,15,10)(8,11)(9,12)(14,18,16,20)(17,19);(1,2,3,4)(5,7)(6,10)(8,16)(9,15)(11,18)(12,13)(14,17)(19,20);(2,4)(5,8)(6,11)(7,14,15,16)(9,17)(10,18,13,20)(12,19);(11,19)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237999, #72238596, #72240429, … |
| maximal normal subgroup of | #72237999, #72238596, #72240429, … |
| maximal quotient of | #72267759 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379721);