Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1304 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| socle | C2 | C2×#1304 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1376 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,13)(4,9)(5,11,6,10)(7,12)(8,15,22,20)(14,27)(16,23)(17,25,18,24)(19,26)(21,28,31,30)(29,32);(1,3,6,15,7,16,5,20)(2,8,11,23,12,22,10,13)(4,14,18,28,19,29,17,30)(9,21,25,32,26,31,24,27);(1,4,7,19)(2,9,12,26)(3,14,16,29)(5,17,6,18)(8,21,22,31)(10,24,11,25)(13,27,23,32)(15,28,20,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240480, #72240481, #72240483, … |
| maximal normal subgroup of | #72240480, #72240481, #72240483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35216);