Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2842 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14673 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,19)(6,10)(7,13)(8,17)(9,21)(12,27)(14,23)(16,24)(18,25)(20,26)(22,28)(29,31)(30,32);(1,3)(2,7,10,11)(4,16)(5,20)(6,13)(8,24,15,29)(9,26,19,23)(12,18)(14,21)(17,31)(22,30,27,25)(28,32);(1,4,5,18,6,17,21,32)(2,8,9,25,10,15,19,30)(3,12,14,16,13,28,20,31)(7,22,23,24,11,27,26,29);(12,28)(16,31)(22,27)(24,29)Polycyclic presentation:
2^9|b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251917, #72251921, #72251929, … |
| maximal normal subgroup of | #72251917, #72251921, #72251929, … |
| maximal quotient of | #72266601 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299229);