Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508062 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 128 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:
(1,2,9,23)(3,13,24,6)(4,14,20,31)(5,12,8,7)(10,16,22,15)(11,26,29,17)(18,30,21,27)(19,32,28,25);(1,3,9,24)(2,10,23,22)(4,17,20,26)(5,15,8,16)(6,12,13,7)(11,27,29,30)(14,28,31,19)(18,32,21,25);(1,4)(2,11)(3,14)(5,18)(6,17)(7,19)(8,21)(9,20)(10,25)(12,28)(13,26)(15,30)(16,27)(22,32)(23,29)(24,31);(4,18)(11,28)(14,30)(17,32)(19,29)(20,21)(25,26)(27,31)Polycyclic presentation:
2^9|a^2=ei,b^2=ei,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72233413, #72233442, #72233448, … |
| maximal normal subgroup of | #72233413, #72233442, #72233448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415334);