Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493638 |
| 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 | #63045 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2332 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)(7,14,20,26)(12,24)(13,27,25,23)(16,22)(21,28)(29,32)(30,31);(1,3)(2,7)(4,13)(5,14)(6,12)(8,21)(9,11)(10,20)(15,29)(16,28)(17,25)(18,24)(19,26)(22,23)(27,30)(31,32);(3,12)(7,20)(9,18)(13,25)(14,26)(21,29)(22,30)(28,32);(1,4)(2,8)(3,13)(5,16)(6,17)(7,21)(9,22)(10,15)(11,23)(12,25)(14,28)(18,30)(19,31)(20,29)(24,27)(26,32)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72225826, #72232878, #72233206, … |
| maximal normal subgroup of | #72225826, #72232878, #72233206, … |
| maximal quotient of | #72225802, #72226081, #72232823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400834);