Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #357 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2)(3,4)(5,6,8,11,9,12,7,10)(14,20,21,25)(15,16)(17,33)(18,34)(22,40,36,35)(23,42,38,41)(24,43,39,37)(26,29)(27,44)(28,32)(30,31);(2,4)(6,10,12,11)(13,14,16,22,29,38,27,43,34,39,44,42,33,40,15,25,19,21,32,36,17,23,30,37,18,24,31,41,26,35,28,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fgi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286481, #72286684, #72286748, … |
| maximal normal subgroup of | #72286481, #72286684, #72286748, … |
| maximal quotient of | #72286519, #72321145 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1247);