Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,14,17,27)(4,8,15,26)(5,13)(6,21,19,30)(7,12)(9,18,28,32)(10,20)(11,31,24,23)(16,29,22,25);(1,3,5,4)(2,8,10,27)(6,22,19,32)(7,17,20,15)(9,11,28,30)(12,26,13,14)(16,23,18,31)(21,25,24,29);(1,4)(2,9,13,25)(3,5)(6,18)(7,15)(8,24,14,11)(10,28,12,29)(16,19)(17,20)(21,26,30,27)(22,23)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gh,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,f^c=fh,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72236605, #72236735, #72236741, … |
| maximal normal subgroup of | #72236605, #72236735, #72236741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47015);