Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11499 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1107 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1107 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,5,10)(3,11,13,25,14,7,17,20)(4,8,16,21)(6,9,18,23)(12,19,27,30,26,24,29,32)(15,22,28,31);(1,3)(2,7)(4,12)(5,17)(6,14)(8,19)(9,11)(10,20)(13,18)(15,26)(16,29)(21,30)(22,24)(23,25)(27,28)(31,32);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,24)(13,27)(14,26)(17,29)(18,28)(20,30)(23,31)(25,32)Polycyclic presentation:
2^9|a^2=e,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,e^b=ef,g^b=gi,d^c=di,e^d=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237425, #72237721, #72237723, … |
| maximal normal subgroup of | #72237425, #72237721, #72237723, … |
| maximal quotient of | #72338752, #72340885, #72340889 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56937);