Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116028 |
| 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 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,22)(3,12)(4,16)(8,24,26,21)(9,13,11,23)(14,31)(15,29)(17,18)(20,27,30,25)(28,32);(2,8,9,25)(3,12)(4,17)(7,20,13,24)(10,26,11,27)(14,32)(15,29)(16,18)(21,22,30,23)(28,31);(1,2)(3,13)(4,8)(5,9)(6,11)(7,15)(10,19)(12,22)(14,30)(16,26)(17,25)(18,27)(20,32)(21,31)(23,29)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237938, #72237987, #72237990, … |
| maximal normal subgroup of | #72237938, #72237987, #72237990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23224);