Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185273 |
| 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 | #62819 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,11,24)(4,17)(5,19)(9,25,28,26)(10,23,12,20)(13,14)(18,32)(22,29,31,27);(1,2)(3,9)(4,8)(5,20)(6,10)(7,12)(11,21)(13,26)(14,31)(15,27)(16,29)(17,23)(18,22)(19,24)(25,32)(28,30);(1,3,6,15)(2,9,10,27)(4,18,17,32)(5,13,19,14)(7,16,21,30)(8,22,23,25)(11,28,12,29)(20,26,24,31);(1,4,7,5)(2,8,12,20)(3,13,16,18)(6,17,21,19)(9,26,29,22)(10,23,11,24)(14,30,32,15)(25,27,31,28)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268067, #72268071, #72268103, … |
| maximal normal subgroup of | #72268067, #72268071, #72268103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92473);