Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10337 |
| 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 | |
772 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 772 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)(3,12,15,23)(4,16)(5,10)(6,20)(7,11)(8,14,24,19)(9,17)(13,28,30,22)(18,25)(21,26)(27,31,32,29);(1,3)(2,8)(4,13)(5,19)(6,14)(7,15)(9,22)(10,23)(11,24)(12,20)(16,32)(17,31)(18,30)(21,29)(25,27)(26,28);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,31)(15,30)(16,20)(19,29)(23,32)(24,28)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,c^b=cf,d^b=di,e^b=eh,g^b=gi,d^c=dg,e^c=ei,h^c=hi,f^d=fi,f^e=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72210561, #72213574, #72283320, … |
| maximal normal subgroup of | #72210561, #72213574, #72283320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51596);