Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1889 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,9,13,22)(5,10)(7,17)(8,20)(14,27)(15,24)(16,30,19,29)(18,26)(21,28)(23,32,25,31);(1,3,13,15,6,14,4,17)(2,7,22,12,11,24,9,27)(5,19,21,32,20,16,18,31)(8,23,26,29,10,25,28,30);(2,8)(5,20)(7,23)(9,28)(10,11)(12,29)(16,19)(18,21)(22,26)(24,25)(27,30)(31,32);(3,14)(7,24)(12,27)(15,17)(16,19)(23,25)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236949, #72237532, #72237535, … |
| maximal normal subgroup of | #72236949, #72237532, #72237535, … |
| maximal quotient of | #72267960, #72312068, C2×#494609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401805);