Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3698 conjugacy classes of subgroups; only the 498 normal subgroups are shown.
This group has 3698 conjugacy classes of subgroups — too many to draw. Only its 498 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) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,18,20,21)(19,23,24,22)(25,26)(27,29)(28,31)(30,32);(2,5)(6,8)(18,21)(19,24)(26,27)(28,32);(3,7)(6,8)(18,21)(22,23)(25,26,29,27)(28,31,32,30);(1,2)(3,8)(4,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,26)(28,30,32,31);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,21)(22,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227426 |
| maximal subgroup of | #72227426, #72227435, #72327832, … |
| maximal normal subgroup of | #72227426, #72227435, #72327832, … |
| maximal quotient of | #72227424, #72268609, #72329174 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028735);