Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484181 |
| 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×C4⋊C4 | C24 |
| prime radical | #12096 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
850 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)(20,22)(21,25)(24,28)(26,31)(27,29)(30,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(20,28)(27,32);(18,22)(19,26)(20,28)(23,30)(24,25)(29,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,21,26)(18,23,25,31)(20,27,28,32)(22,29,24,30)Polycyclic presentation:
2^9|d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72318727 |
| maximal normal subgroup of | #72318727 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391714);