Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485320 |
| 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 | |
1259 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)(3,13)(4,11)(5,18)(6,10)(7,12)(8,14)(9,17)(15,29)(16,28)(19,24)(20,27)(21,26)(22,31)(23,32)(25,30);(1,3)(2,8)(4,14)(5,19)(6,16)(7,15)(9,22)(10,24)(11,28)(12,25)(13,18)(17,30)(20,32)(21,31)(23,26)(27,29);(1,4,6,5)(2,9,26,18)(3,14,16,19)(7,17,21,20)(8,22,23,13)(10,27,12,11)(15,30,31,32)(24,29,25,28);(2,10)(3,15)(8,23)(9,27)(11,18)(12,26)(13,22)(14,30)(16,31)(19,32)(24,25)(28,29)Polycyclic presentation:
2^9|c^2=gh,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,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72285914 |
| maximal normal subgroup of | #72285914 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392589);