Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #340 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C42×C8 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 512 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6)(3,4)(7,8)(9,10,11,13,12,14,15,16)(17,18,20,25,22,27,21,26)(19,28,29,32,31,23,30,24);(1,2,4,7,5,8,3,6)(18,23)(19,29)(20,21)(24,26)(25,32)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216287, #72216338, #72216346, … |
| maximal normal subgroup of | #72216287, #72216338, #72216346, … |
| maximal quotient of | #72216280, #72216557, #72225347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 654);