Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| socle | C2 | C22×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,9)(5,10)(6,11)(7,14)(12,21)(13,22)(15,23)(16,24)(17,19)(18,25)(20,27)(26,31)(28,30)(29,32);(1,3)(2,7)(4,13)(5,14)(6,12)(8,10)(9,20)(11,19)(15,27)(16,26)(17,18)(21,25)(22,23)(24,30)(28,29)(31,32);(1,3)(2,8)(4,13)(5,17)(6,12)(7,10)(9,22)(11,21)(14,18)(15,28)(16,26)(19,25)(20,23)(24,31)(27,29)(30,32);(3,12)(7,19)(8,21)(13,26)(14,17)(20,30)(22,31)(27,28);(1,4,6,16)(2,9,11,24)(3,13,12,26)(5,15,18,29)(7,20,19,30)(8,22,21,31)(10,23,25,32)(14,27,17,28)Polycyclic presentation:
2^9|e^2=i,b^a=bf,c^a=cg,h^a=hi,d^b=di,g^b=ghi,d^c=di,f^c=fh,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236900, #72236922, #72237528, … |
| maximal normal subgroup of | #72236900, #72237528, #72237532, … |
| maximal quotient of | #72256207, #72256210, #72262412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530070);