Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116392 |
| 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 | |
803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,23)(3,12)(4,16)(8,24,26,22)(9,21,11,14)(13,28)(15,30)(17,18)(20,27,32,25)(29,31);(2,8,9,25)(3,13)(4,17)(7,20,21,24)(10,26,11,27)(12,28)(14,22,23,32)(15,31)(16,18)(29,30);(1,2)(3,14)(4,8)(5,9)(6,11)(7,13)(10,19)(12,23)(15,20)(16,26)(17,25)(18,27)(21,28)(22,30)(24,29)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ei,g^c=ghi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233090, #72233128, #72233134, … |
| maximal normal subgroup of | #72233090, #72233128, #72233134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23588);