Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(17,18,22,24,21,20,26,29)(19,25,28,32,27,23,30,31);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,26,28,21,27,22,30)(18,23,24,32,20,25,29,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,20)(18,21)(19,25)(22,24)(23,27)(26,29)(28,31)(30,32);(18,20)(19,27)(24,29)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#196224, #72215940, #72215973, … |
| maximal normal subgroup of | C2×#196224, #72215940, #72215973, … |
| maximal quotient of | #72215865, C2×#196224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103420);