Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61431 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,10,14)(4,6,12,9)(11,13,16,15)(17,18,21,25,22,19,26,24)(20,27,28,31,29,23,30,32);(2,6)(4,12)(7,9)(8,13)(11,16)(14,15)(17,19)(18,22)(20,27)(21,24)(23,29)(25,26)(28,31)(30,32);(2,7)(6,9)(8,14)(13,15)(17,20,26,30,22,29,21,28)(18,23,24,31,19,27,25,32);(1,3)(2,8)(4,11)(5,10)(6,13)(7,14)(9,15)(12,16);(3,10)(8,14)(11,16)(13,15)Polycyclic presentation:
2^9|a^2=g,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=ehi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72216509, #72216771, #72216779, … |
| maximal normal subgroup of | #72216509, #72216771, #72216779, … |
| maximal quotient of | D4×#1580, #72224967, C2×#6345865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253061);