Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2686 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15475 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
877 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)(3,15)(4,19,23,10)(5,11)(6,14)(7,12)(8,13)(9,17)(16,27,28,21)(18,24)(20,26,31,32)(22,29,25,30);(1,3,6,18,8,11,7,17)(2,9,14,15,13,24,12,5)(4,20,22,16,23,31,25,28)(10,27,30,32,19,21,29,26);(1,4)(2,10)(3,16)(5,21)(6,22)(7,25)(8,23)(9,26)(11,28)(12,29)(13,19)(14,30)(15,27)(17,20)(18,31)(24,32);(1,5,6,9,8,15,7,24)(2,11,12,18,13,3,14,17)(4,21,22,26,23,27,25,32)(10,28,29,31,19,16,30,20)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319935, #72319947, #72320069, … |
| maximal normal subgroup of | #72319935, #72319947, #72320069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340422);