Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493635 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #63017 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2290 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)(3,11)(4,15,17,8)(5,18,19,9)(6,10)(7,13,21,28)(12,25,27,24)(14,23)(16,22)(20,26)(29,32)(30,31);(1,3,6,14)(2,7,10,21)(4,12,17,27)(5,13,19,28)(8,20,15,29)(9,11,18,23)(16,26,31,32)(22,24,30,25);(9,18)(11,23)(22,30)(24,25);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,24)(13,26)(14,27)(18,30)(19,31)(21,29)(23,25)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232878, #72233206, #72240644, … |
| maximal normal subgroup of | #72232878, #72233206, #72240644, … |
| maximal quotient of | #72232764, #72233201, #72233339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400832);