Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417669 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15369 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1225 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)(3,7)(4,16,18,28)(5,10)(6,11)(8,20,27,19)(9,22)(12,15)(13,30,17,29)(14,24)(21,25)(23,32,26,31);(1,3)(2,7)(4,17)(5,21)(6,14)(8,26)(9,12)(10,25)(11,24)(13,18)(15,22)(16,30)(19,31)(20,32)(23,27)(28,29);(1,2)(3,12)(4,8)(5,10)(6,11)(7,21)(9,22)(13,29)(14,25)(15,24)(16,20)(17,30)(18,27)(19,28)(23,32)(26,31);(1,4)(2,8)(3,13,14,17)(5,19)(6,18)(7,23,24,26)(9,16)(10,28)(11,27)(12,29,25,30)(15,31,21,32)(20,22)Polycyclic presentation:
2^9|a^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319162, #72319370, #72319406, … |
| maximal normal subgroup of | #72319162, #72319370, #72319406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334572);