Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487635 |
| 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 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1080 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)(3,12)(4,7)(5,10)(6,11)(8,14)(9,16)(13,27)(15,22)(17,20)(18,25)(19,30)(21,31)(23,28)(24,32)(26,29);(2,7)(3,13)(4,16)(8,21)(9,10)(11,20)(12,26)(14,28)(15,31)(17,32)(19,23)(22,27)(24,25)(29,30);(3,14)(4,17)(8,12)(13,28)(15,31)(16,32)(19,22)(21,26)(23,27)(29,30);(1,3)(2,8)(4,15)(5,14)(6,13)(7,19)(9,22)(10,12)(11,23)(16,30)(17,29)(18,28)(20,21)(24,26)(25,27)(31,32)Polycyclic presentation:
2^9|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231830, #72283025, #72285284, … |
| maximal normal subgroup of | #72231830, #72283025, #72285284, … |
| maximal quotient of | #72284486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394831);