Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2517 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16429 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,25,7,13,6,12)(3,14,23,9,16,32,15,29)(4,17,18,28,19,10,26,30)(5,20,21,27,22,11,24,31);(1,3)(2,9)(4,5)(6,23)(7,16)(8,15)(10,27)(11,28)(12,32)(13,29)(14,25)(17,31)(18,24)(19,22)(20,30)(21,26);(1,4,6,18,7,19,8,26)(2,10,12,30,13,17,25,28)(3,5,15,21,16,22,23,24)(9,27,14,11,29,31,32,20);(1,5)(2,11,13,20)(3,4)(6,24)(7,22)(8,21)(9,28,29,30)(10,14,17,32)(12,27,25,31)(15,26)(16,19)(18,23)Polycyclic presentation:
2^9|a^2=eg,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319859, #72319935, #72319988, … |
| maximal normal subgroup of | #72319859, #72319935, #72319988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369588);