Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62818 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,20)(4,15)(8,19,24,31)(9,16,11,21)(13,25)(17,30)(22,32,23,27)(28,29);(1,2)(3,8)(4,16)(5,11)(6,9)(7,17)(10,18)(12,22)(13,27)(14,23)(15,20)(19,29)(21,30)(24,26)(25,31)(28,32);(3,12)(4,17)(7,16)(8,22)(13,28)(14,26)(15,30)(19,31)(20,21)(23,24)(25,29)(27,32);(1,3,6,14)(2,8,9,23)(4,13,17,29)(5,12,18,26)(7,19,16,27)(10,24,11,22)(15,25,30,28)(20,31,21,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232836, #72233035, #72233125, … |
| maximal normal subgroup of | #72232836, #72233035, #72233125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94020);