Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238377 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,22)(3,11)(4,16)(8,25,27,19)(9,21,23,12)(14,30)(15,20)(17,18)(24,26,29,32)(28,31);(2,8,10,27)(3,11)(4,17)(7,19,22,25)(9,26,23,32)(12,29,21,24)(14,31)(15,20)(16,18)(28,30);(1,2)(3,12)(4,8)(5,10)(6,9)(7,20)(11,22)(13,23)(14,25)(15,21)(16,32)(17,27)(18,26)(19,30)(24,31)(28,29);(2,9)(7,21)(8,26)(10,23)(12,22)(19,24)(25,29)(27,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fhi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233090, #72233129, #72233133, … |
| maximal normal subgroup of | #72233090, #72233129, #72233133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145573);