Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116129 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,21)(3,11)(8,20,25,32)(9,19,24,31)(12,26)(13,27)(15,30,23,18)(28,29);(2,8,9,23)(3,12)(4,14)(7,18,19,20)(10,25,24,15)(11,26)(13,29)(17,22)(21,30,31,32)(27,28);(1,2)(3,7)(4,15)(5,9)(6,10)(8,22)(11,21)(12,19)(13,30)(14,23)(16,24)(17,25)(18,27)(20,29)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,f^2=hi,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,f^c=fhi,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233090, #72233126, #72233130, … |
| maximal normal subgroup of | #72233090, #72233126, #72233130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23325);