Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351869 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2915 conjugacy classes of subgroups; only the 566 normal subgroups are shown.
This group has 2915 conjugacy classes of subgroups — too many to draw. Only its 566 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) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,16)(18,23)(20,21)(24,31)(25,26)(27,28)(30,32);(1,2,6,9)(3,8,13,15)(4,10,5,7)(11,16,12,14)(17,18,22,26)(19,24,29,32)(20,23,21,25)(27,30,28,31);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32);(2,9)(7,10)(8,15)(14,16)(17,20,22,21)(18,25,26,23)(19,27,29,28)(24,31,32,30);(1,4,6,5)(2,10,9,7)(3,12,13,11)(8,14,15,16)(17,20,22,21)(18,23,26,25)(19,28,29,27)(24,31,32,30)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
| Fitting subgroup of | #72272596 |
| maximal subgroup of | #72272561, #72272568, #72272596, … |
| maximal normal subgroup of | #72272561, #72272568, #72272596, … |
| maximal quotient of | #72341054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259078);