Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1387 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16)(8,13,22,11)(9,23)(10,15)(12,19)(14,25)(17,32)(18,31,20,36)(21,27,24,30)(26,35,28,34)(29,33);(1,2,3,4)(6,11,15,13)(7,17)(12,28,29,26)(14,30,23,27)(16,31,32,36)(18,20)(19,33)(34,35);(5,7,9,19,10,17,25,33)(6,12,14,16,15,29,23,32)(8,20,24,34,22,18,21,35)(11,26,30,36,13,28,27,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312053, #72312068, #72312082, … |
| maximal normal subgroup of | #72312053, #72312068, #72312082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54691);