Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144380 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10336 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23895 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
933 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 933 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 3 generators on 32 points:
(1,2)(3,8)(4,15)(5,20)(6,10)(7,11)(9,17)(12,25)(13,27)(14,24)(16,22)(18,26)(19,28)(21,23)(29,32)(30,31);(1,3)(2,8)(4,16)(5,13)(6,21)(7,14)(9,25)(10,27)(11,24)(12,18)(15,22)(17,29)(19,30)(20,23)(26,32)(28,31);(1,4,7,19)(2,9,11,26)(3,12,14,29)(5,18,6,17)(8,22,24,31)(10,28,20,15)(13,30,21,16)(23,32,27,25)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72236674, #72251567, #72258232, … |
| maximal normal subgroup of | #72236674, #72251567, #72258232, … |
| maximal quotient of | #72236911, #72258405 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51577);