Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511816 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2385 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2385 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,10)(5,18)(6,9)(7,12)(8,15)(11,21)(14,29)(16,28)(17,26)(19,25)(20,27)(22,30)(23,32)(24,31);(1,3)(2,8)(4,14)(5,19)(6,22)(7,16)(9,23)(10,24)(11,28)(12,25)(13,18)(15,21)(17,32)(20,31)(26,29)(27,30);(2,9)(4,6)(8,23)(10,12)(11,27)(13,29)(14,22)(17,20)(18,26)(24,25)(28,30)(31,32);(1,4,7,6)(2,10,12,9)(3,14,16,22)(5,20,21,17)(8,24,25,23)(11,26,18,27)(13,30,28,29)(15,32,19,31)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72234207, #72260900, #72261107, … |
| maximal normal subgroup of | #72234207, #72260900, #72261107, … |
| maximal quotient of | #72286559, #72286626 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419013);