Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| 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 | #61432 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1579 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)(3,8)(4,6)(5,7)(9,12)(10,14)(11,13)(15,16)(17,18,21,25,22,19,26,24)(20,23,28,31,29,27,32,30);(2,6)(4,12)(7,9)(8,13)(11,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,31)(30,32);(2,7)(6,9)(8,14)(13,15)(17,20,21,28,22,29,26,32)(18,23,25,31,19,27,24,30);(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=g,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,c^b=ci,f^b=fh,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72338444, #72338445, #72338446, … |
| maximal normal subgroup of | #72338444, #72338445, #72338446, … |
| maximal quotient of | #72343877, #72343878, #72343882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253173);