Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493397 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3483 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)(3,11,14,25)(4,8)(5,17,20,9)(6,10)(7,13)(12,28,29,27)(15,23)(16,26,19,24)(18,22)(21,30)(31,32);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,25)(10,22)(11,17)(13,20)(15,29)(16,31)(19,30)(23,32)(24,27)(26,28);(1,4)(2,8)(3,12)(5,19)(6,15)(7,21)(9,26)(10,23)(11,27)(13,31)(14,29)(16,20)(17,24)(18,30)(22,32)(25,28);(4,15)(8,23)(12,29)(16,19)(21,32)(24,26)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,e^c=ei,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213435, #72213504, #72213523, … |
| maximal normal subgroup of | #72213435, #72213504, #72213523, … |
| maximal quotient of | #72232763, #72233189, #72233270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400593);