Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486803 |
| 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 | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,7)(5,10)(6,11)(9,16)(12,24)(13,21)(14,22)(15,18)(17,19)(20,27)(23,28)(25,29)(26,32)(30,31);(2,7)(3,12)(4,16)(8,20)(9,10)(11,19)(13,26)(15,30)(17,29)(18,24)(21,31)(22,25)(23,32)(27,28);(3,13)(4,17)(8,21)(12,26)(15,30)(16,29)(18,23)(20,31)(24,32)(27,28);(1,3,14,26)(2,8,22,32)(4,15,29,30)(5,13,6,12)(7,18,25,31)(9,23,19,20)(10,21,11,24)(16,28,17,27)Polycyclic presentation:
2^9|d^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72231386, #72234735, #72234738, … |
| maximal normal subgroup of | #72231386, #72234735, #72234738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394544);