Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15561 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 180 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)(3,7,13,21)(4,8)(5,17,18,9)(6,10)(11,22,14,19)(12,25)(15,29,30,23)(16,24)(20,27)(26,31)(28,32);(1,3)(2,7,10,21)(4,14,16,11)(5,12,18,27)(6,13)(8,22)(9,25)(15,26)(17,20)(19,24)(23,32,29,31)(28,30);(1,4)(2,8)(3,11)(5,15)(6,16)(7,19)(9,23)(10,24)(12,26)(13,14)(17,29)(18,30)(20,31)(21,22)(25,32)(27,28);(1,5)(2,9)(3,12,13,27)(4,15)(6,18)(7,20,21,25)(8,23)(10,17)(11,26,14,28)(16,30)(19,31,22,32)(24,29)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237722, #72237724, #72237746, … |
| maximal normal subgroup of | #72237722, #72237724, #72237746, … |
| maximal quotient of | #72265874 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343660);