Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2815 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16252 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 1938 conjugacy classes of subgroups — too many to draw. Only its 216 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)(3,14)(4,9)(5,12)(6,11)(7,13)(8,16)(10,18)(15,27)(17,25)(19,24)(20,31)(21,29)(22,30)(23,28)(26,32);(1,3,5,16,7,17,6,18)(2,8,11,14,13,10,12,25)(4,19,21,32,22,20,23,15)(9,26,28,24,30,27,29,31);(1,4)(2,9)(3,15)(5,23)(6,21)(7,22)(8,24)(10,31)(11,29)(12,28)(13,30)(14,26)(16,20)(17,32)(18,19)(25,27);(1,3,5,16,7,17,6,18)(2,10,11,25,13,8,12,14)(4,20,21,15,22,19,23,32)(9,27,28,31,30,26,29,24)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216123, #72216320, … |
| maximal normal subgroup of | #72216037, #72216123, #72216320, … |
| maximal quotient of | C2×#454389, #72239650, #72255629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361585);