Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #36714 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10)(3,11,14,25)(5,9,17,24)(6,8,16,23)(7,15,20,13)(12,19,27,30)(18,22,29,32)(21,28,31,26);(1,3)(2,7)(4,15)(5,13)(6,12)(8,21)(9,11)(10,25)(14,17)(16,28)(18,26)(19,22)(20,24)(23,30)(27,29)(31,32);(2,8)(4,16)(7,19)(9,22)(11,21)(14,27)(15,28)(17,29)(20,25)(30,31);(2,9)(3,12)(4,17)(7,19)(8,22)(11,21)(13,26)(14,28)(15,27)(16,29)(20,31)(25,30)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72240624, #72240627, #72240629, … |
| maximal normal subgroup of | #72240624, #72240627, #72240629, … |
| maximal quotient of | #72251430 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121072);