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 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 1089 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,7)(5,9)(8,13)(10,14)(11,15)(12,16);(1,3)(2,6)(4,10)(5,12)(7,15)(8,14)(9,16)(11,13);(2,7)(3,10)(4,13)(8,9)(11,12)(14,15)Polycyclic presentation:
2^9|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 | #72210342, #72210561, #72213574, … |
| maximal normal subgroup of | #72210342, #72210561, #72213574, … |
| maximal quotient of | #72210569, #72236913, C2×#144380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51576);