Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486803 |
| 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 |
| prime radical | #29686 | C2 |
| 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 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,23)(20,28)(21,25)(22,27)(24,30)(26,32)(29,31);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,22)(20,29)(21,26)(23,28)(24,32)(25,31)(27,30);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,19)(18,23)(20,29)(21,26)(22,28)(24,31)(25,32)(27,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,26)(22,31)(23,32)(27,29)Polycyclic presentation:
2^9|b^2=gi,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 | #72341194, #72341202 |
| maximal normal subgroup of | #72341202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394628);