Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622880 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,14)(10,12)(11,13)(15,16);(2,6)(7,12)(8,13)(14,16);(4,11)(10,15);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16);(3,9)(7,14)(10,15)(12,16)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,h^a=hi,g^b=ghi,f^c=fh,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213368, #72216366, #72217027, … |
| maximal normal subgroup of | #72213368, #72216366, #72233206, … |
| maximal quotient of | #72210487, #72213210, #72232895, … |
| automorphism group of | #410, #413 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530076);