Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17105 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1169 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 1169 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,16,9,17,25,22,24)(5,18,19,27,21,14,20,10)(15,26,30,28,32,23,31,29);(1,3,6,11,8,12,7,2)(4,9,16,13,17,24,22,25)(5,10,19,14,21,27,20,18)(15,29,30,23,32,28,31,26);(1,4)(2,9,11,24)(3,13,12,25)(5,15)(6,16)(7,22)(8,17)(10,26,27,23)(14,29,18,28)(19,30)(20,31)(21,32);(1,5)(2,10,11,27)(3,14,12,18)(4,15)(6,19)(7,20)(8,21)(9,23,24,26)(13,28,25,29)(16,30)(17,32)(22,31)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285123, #72285150, #72285220, … |
| maximal normal subgroup of | #72285123, #72285150, #72285220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384371);