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 | #63021 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2263 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)(3,13,16,25)(4,9)(5,10,19,28)(6,11,21,29)(7,12)(8,15)(14,27)(17,26,20,23)(18,22)(24,31)(30,32);(1,3,7,16)(2,8,12,24)(4,5,14,19)(6,15,21,31)(9,22,27,32)(10,23,28,26)(11,13,29,25)(17,18,20,30);(1,4,7,14)(2,9,12,27)(3,5,16,19)(6,20,21,17)(8,22,24,32)(10,25,28,13)(11,26,29,23)(15,30,31,18);(1,5)(2,10)(3,14)(4,16)(6,18)(7,19)(8,23)(9,25)(11,22)(12,28)(13,27)(15,20)(17,31)(21,30)(24,26)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^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 | #72240644, #72249650, #72249658, … |
| maximal normal subgroup of | #72240644, #72249650, #72249658, … |
| maximal quotient of | #72255684, #72255933, #72262135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400833);