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:
(1,2)(3,5)(4,7)(6,8)(9,10,11,13,12,14,15,16)(18,23,24,27)(19,20)(21,29)(25,32,31,26)(28,30);(2,5)(3,8)(6,7)(17,18,20,25,29,32,19,27,22,24,30,31,21,26,28,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=fg,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216306, #72216313, #72216346, … |
| maximal normal subgroup of | #72216306, #72216313, #72216346, … |
| maximal quotient of | #72216302, #72216557, #72225349 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 648);