Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479301 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17198 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 799 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,7,12,19)(4,14,15,29)(5,10)(6,9)(8,16,23,30)(11,26)(13,21,17,25)(18,24)(20,31)(22,28)(27,32);(1,3)(2,7,9,19)(4,11)(5,17)(6,12)(8,22,23,20)(10,25,24,21)(13,18)(14,26,29,32)(15,27)(16,31)(28,30);(5,18)(7,19)(8,23)(10,24)(11,27)(13,17)(16,30)(20,22);(1,4)(2,8,9,23)(3,11)(5,16)(6,15)(7,20,19,22)(10,14,24,29)(12,27)(13,28)(17,31)(18,30)(21,26,25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285022, #72285055, #72285868, … |
| maximal normal subgroup of | #72285022, #72285055, #72285868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386497);