Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
1012 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)(12,20)(13,21)(14,18)(15,23)(17,19)(22,25)(24,27)(26,31)(28,32)(29,30);(2,7)(4,16)(8,18)(9,10)(11,19)(14,25)(17,27)(20,22)(21,24)(23,30)(28,29)(31,32);(3,12)(4,17)(8,20)(14,28)(15,26)(16,27)(18,22)(23,31)(25,29)(30,32);(1,3,13,26)(2,8,21,31)(4,14,27,28)(5,12,6,15)(7,18,24,32)(9,22,19,30)(10,20,11,23)(16,25,17,29)Polycyclic presentation:
2^9|d^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72318749, #72341202 |
| maximal normal subgroup of | #72318749, #72341202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394497);