Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116183 |
| 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 | |
849 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)(3,11)(4,15)(8,23,27,21)(9,20,26,29)(12,14)(16,32,25,19)(18,24);(2,8,9,25)(3,12)(4,15)(7,19,20,23)(10,27,26,16)(11,14)(13,31)(18,24)(21,22,32,29)(28,30);(1,2)(3,7)(4,16)(5,9)(6,10)(8,24)(11,22)(12,29)(13,19)(14,20)(15,25)(17,26)(18,27)(21,30)(23,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,f^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fi,d^c=dh,f^c=fhi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233090, #72233125, #72233131 |
| maximal normal subgroup of | #72233090, #72233125, #72233131 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23379);