Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #35528 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1397 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1397 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 20 points:
(5,6)(7,15)(8,12,9,13)(10,14)(11,16,19,18)(17,20);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19);(2,4)(6,12)(8,9)(11,15)(13,14)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=ghi,d^c=di,e^c=ei,g^c=ghi,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72234157, #72234167, #72249503, … |
| maximal normal subgroup of | #72234157, #72234167, #72249503, … |
| maximal quotient of | #72234154, #72234549, #72261967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54937);