Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8316 |
| prime radical | #3481 | C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 32 points:
(1,2,4,9,6,10,17,23,5,11,16,24,18,25,29,32)(3,8,13,21,14,22,27,31,12,7,15,19,26,20,28,30);(2,7)(3,12)(8,11)(9,21)(10,20)(13,15)(14,26)(19,24)(22,25)(23,31)(27,28)(30,32);(3,12)(13,15)(14,26)(27,28)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231599, #72231607, #72231612, … |
| maximal normal subgroup of | #72231599, #72231607, #72231612, … |
| maximal quotient of | #72239963, C2×#129615 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36811);