Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459145 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2751 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16368 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,16,8,11,6,12)(3,4,15,19,9,20,14,18)(5,10,21,26,22,25,23,27)(13,17,28,30,24,31,29,32);(1,3,6,15,8,9,7,14)(2,4,16,18,11,20,12,19)(5,13,21,29,22,24,23,28)(10,17,27,30,25,31,26,32);(1,4)(2,9)(3,11)(5,17)(6,19)(7,18)(8,20)(10,13)(12,14)(15,16)(21,32)(22,31)(23,30)(24,25)(26,29)(27,28);(1,5)(2,10)(3,13)(4,17)(6,21)(7,23)(8,22)(9,24)(11,25)(12,26)(14,28)(15,29)(16,27)(18,30)(19,32)(20,31)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284248, #72319935, #72319962, … |
| maximal normal subgroup of | #72284248, #72319935, #72319962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367823);