Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511842 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32926 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1043 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,11)(4,15)(5,8)(6,10)(7,13)(9,17)(12,25)(14,26)(16,30)(18,23)(19,32)(20,27)(21,29)(22,31)(24,28);(1,3,4,13,6,12,17,27)(2,7,9,11,10,20,15,25)(5,14,16,29,18,28,31,32)(8,21,22,26,23,19,30,24);(2,8)(3,12)(7,19)(9,22)(10,23)(11,24)(13,27)(14,28)(15,30)(20,21)(25,26)(29,32);(3,12)(5,18)(7,20)(11,25)(13,27)(16,31)(19,21)(24,26)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^b=di,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72236997, #72237519, #72237535, … |
| maximal normal subgroup of | #72236997, #72237519, #72237535, … |
| maximal quotient of | #72311922, #72312131, #72312168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419128);