Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125642 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7734 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 3 generators on 32 points:
(1,2)(3,12)(4,16)(5,10)(6,11)(7,13)(8,18)(9,19)(14,28)(15,29)(17,22)(20,26)(21,25)(23,30)(24,31)(27,32);(1,3)(2,7)(4,17)(5,21)(6,15)(8,25)(9,22)(10,23)(11,24)(12,16)(13,18)(14,19)(20,30)(26,29)(27,28)(31,32);(4,18)(5,19)(8,26)(9,10)(12,28)(13,17)(14,21)(16,27)(20,32)(22,23)(25,29)(30,31)Polycyclic presentation:
2^9|d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ei38 conjugacy classes — view the full character table.
| maximal subgroup of | #72219621, #72226635, #72226639 |
| maximal normal subgroup of | #72219621, #72226635, #72226639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32838);